Quantum informative analysis for weather of San Francisco

Abstract

We present a novel weather analysis of San Francisco from 2009 to 2019 using concepts from quantum information theory. We describe how to transform a classical weather dataset into a quantum-inspired analytical framework by treating it as a quantum information system embedded in a quantum-like representation, with data encoded as density matrices in Hilbert space. Transforming classical weather data into quantum states is a mathematical abstraction, whereas a quantum-inspired system is not a quantum-physical system. We study a quantum-inspired analytical framework using quantum information-theoretic metrics, including fidelity, classical information, quantum information, decoherent information, Shannon and von Neumann entropies, and the quantum stability index, to represent and analyze the dataset. Furthermore, we give predictions for each quantum information metric for 2020 and 2021. We fit Fourier-series models to the metrics to explore temporal patterns and structural properties in the data.

Introduction

Since the 1990s, quantum information theory has emerged as a new field in physics^1,2. It has become the focus of the attention of many researchers who have begun to solve the problems related to it^1–5. It was developed rapidly and made tremendous progress and was the cornerstone and backbone of the quantum computer^6,7. Quantum information theory has numerous applications in various fields, including the gravitational field, the development of new materials in chemistry, genetics, and others^8–11. In a new topic, the climate is selected as an effective application of quantum information theory. Since the 1980s, climate change has become one of the main problems^12–14. As climate variables change steadily, the climate becomes more violent^15,16. Since the start of 2020, the world has begun a new era of extreme climate change^17,18. Let it serve as a beacon to design mechanisms to understand and comprehend what has happened since 2020 and the emergence of the heat dome phenomenon^19,20. We only had to go back a little to the previous decade, from 2009 to 2019, to learn about the prevailing climate conditions during it²¹.

Thus, quantum information theory, with its numerous tools, is provided as a suitable mathematical model to study and analyze climate data accumulated over previous years²². It has mathematical methods to carry out the required analyses, regardless of the number of variables and the volume of data accumulated²³. Climate data will be examined on multiple scales to help us understand the mechanisms of climate and changes related to it²⁴. Therefore, we will focus on examining climate data for a specific region using its metrics to understand the climate during the entire decade from 2009 to 2019 and to compare the climate conditions of the previous decade with those of the current decade²⁵. It can measure damages caused by climate change and treat their variables without complications²⁶. Therefore, based on its measurements, objective discussions can be conducted and essential results can be drawn to form a future vision on how to confront climate change and find sound solutions to repair the ecosystem in that region²⁷. Consequently, appropriate decisions are taken to find solutions to climate problems²⁸.

In this topic, the manuscript involves the quantum model of classical data, which is quantized as a quantum-inspired analytical framework. In the beginning, there is no actual quantum physical system, but the quantum model is a quantum-inspired analytical framework as a mathematical abstraction. In Sect. "Quantum model of dataset", classical data are modeled within the quantum information system using the quantum informative formalism to obtain a quantum-inspired analytical framework. It is formulated in Hilbert space, which is here named the nine-dimensional feature space to form pure and mixed quantum states. A pure state is in a column matrix and a mixed state is in a square matrix where the matrix form is here suitable for quantum states. In addition, quantum information-theoretic metrics are introduced, such as fidelity, classical, quantum, decoherent information, Shannon entropy, von Neumann entropy, and the quantum stability index. From the viewpoint of Sects. " Quantum model of dataset", "San Francisco weather dataset" deals with the San Francisco weather dataset. Firstly, the dataset is manipulated by a maximal reference to be a dimensionless dataset. Secondly, the data are normalized to obtain a quantum representation. Thirdly, they are transformed into quantum-inspired representations. Fourthly, the data are represented using both pure and mixed quantum states. In Sect. "Discussion and results", these quantum states are analyzed through quantum information-theoretic metrics introduced in Sect. "Quantum model of dataset". This approach leverages the power of quantum metrics to gain new insights and computational advantages beyond traditional analytical methods. Furthermore, Fourier series are employed to fit these matrices, enabling the examination of temporal patterns and structural properties of the data, as well as the prediction of their values for the years 2020 and 2021.

Quantum model of dataset

Consider the classical data involving n rows and d columns provided by the dataset and each data is represented by the numerical value and . Thus, the dataset is written as an rectangular matrix according to the classical information^29–31. This matrix representation is inadequate from a quantum mechanical perspective^22,23. Therefore, the dataset is reformulated in a form accepted in quantum information theory^7,32–34. The dataset is modeled in quantum view as the quantum feature space ^35–39,41. A quantum-inspired framework system is formed as the dimensional quantum feature space of the dataset ^32,35,40,41. In fact, the quantum feature space is taken as the Hilbert space in the quantum framework³⁵. Subsequently, each feature corresponds to a column and is obtained as the standard basis vector in the space ⁴¹. Currently, this quantum framework can be used to establish a quantum-inspired framework system in the space ⁴¹. Consequently, the space is constructed from standard bases , where the basis represents feature; ^35,42. Hence, in the quantum view, each row in is formed by an arbitrary non-normalized vector Therefore, an arbitrary non-normalized vector is written in terms of standard bases as:

1

Hence, the vector is normalized if it is in the form as follows⁴¹:

2

where and In fact, the dataset is a rectangular matrix, but quantum mechanics does not directly treat it. In this respect, a new quantum density state is defined for to obey quantum mechanics. The quantum state is given by the square matrix of order and is introduced as⁴¹:

3

where refers to the trace of the matrix and is normalized. Generally, the state is obtained in terms of standard bases as follows^32,35,41:

4

where such that and

The state is always a mixed state and has a noise that describes the randomness of the data. The classical information consists of diagonal elements and the quantum information represents non-diagonal elements where and . Clearly, a non-diagonal element is the joint information of ket basis and bra basis

In this manuscript, we study the dataset using some quantum informative relations, such as fidelity, classical information CI, quantum information QI, decoherent information DI, quantum stability index Shannon entropy of CI, Shannon entropy of QI, and von Neumann entropy of DI, which are provided later.

If a pure state represents the row and the mixed state is denoted by then the fidelity is defined as³³:

5

where . If , then the state is accepted and otherwise is rejected.

There is an important property of the state if is a pure state, then , and if is a mixed state, then . Based on this property, is carefully investigated. The term is given in terms of state by³³:

6

Equation (6) involves two parts of different physical meaning. The first part represents diagonal elements, which refer to the classical information CI. It is written as:

7

The second part refers to the quantum information QI that describes non-diagonal elements. It is written as follows:

8

Thus, is the sum of two types of information: CI and QI : 

9

Currently, a new type of information must be introduced here. This type of information is decoherent DI. It is known as the purity or linear entropy. It is expressed for the unmeasurable information directly by elements of the state such as CI and QI. It is possible to measure from the following relationship as²⁴:

10

The decoherent information appears to be just mixed, but in the pure state, it vanishes. Subsequently, this quantum information system encompasses three distinct types of information. All types of information CI, QI and DI are included in this relation:

11

Let and We introduce a new measure as the ratio between the decoherent information () and the sum of the classical and quantum information as follows:

12

where is the quantum stability index for the quantum information system and equals^27,28:

13

where Eq. (13) has for four cases. The first case is where i.e., and Then is a pure state and completely stable. The second case is where i.e., Thus, and In this case, is stable and has low noise. The third case is where i.e., Thus, and is metastable. The fourth case is where i.e., . Thus, and Therefore, is unstable.

In this regard, randomness expresses the discrepancy between some data values that appear due to the lack of harmony and compatibility within the information system. Each type of information has a different type of randomness.

As data continues to accumulate within an information system, randomness appears in many forms. Therefore, it is essential to search for valid relations for measuring each type of randomness. Entropy is the most accurate and precise measure for computing randomness. We define three types of information entropy.

Shannon entropy of CI is defined as⁴³:

14

where are diagonal elements of the state and . Shannon entropy of QI is defined as:

15

where points to non-diagonal elements which are ordered from to , the number of non-diagonal terms is , and . measures only the noise of non-diagonal elements. Since the base of the logarithm is d, and whenever the number of nonzero terms exceeds d, we have , where this case can be satisfied when . If the number of nonzero elements does not exceed d, then ranges between 0 and 1. If , then is enclosed between 0 and 1. If the state is described by a diagonal matrix, then all nondiagonal elements are zero and therefore, becomes 0, where by convention.

Von Neumann entropy of DI is defined as follows³³:

16

where is the eigenvalue of the state and If is a pure state, then while if is a fully mixed state. Thus, the randomness of CI, QI, and DI are measured by the entropies and , respectively.

Note that and are determined directly by diagonal elements and non-diagonal elements, respectively, but the eigenvalues of state are computed using the entropy of .

San Francisco weather dataset

The real model is the San Francisco weather dataset, USA, from January 2009 to December 31, 2019²¹. The dataset includes nine weather variables. The weather variables are maximum temperature (in Kelvin K), minimum temperature (K), dew point (K), cloud cover (%), humidity (%), precipitation (mm), pressure (mbar), degree of wind direction, and wind speed (km/h), respectively.

The climate is investigated as a new application of quantum information theory. The San Francisco weather dataset is used as an example. This statistic is later verified by quantum information tools. In this paper, we propose a novel technique for investigating the dynamics of the weather based on quantum information theory as a good exercise.

We propose a quantum-inspired framework that models the San Francisco weather dataset, denoted by , as a quantum information system. Although this system does not represent a physical quantum system, it adopts principles from quantum theory to enable novel data representations and analytical capabilities. The transformation process involves four key steps: first, the maximum values of the weather variables are identified to serve as reference points, allowing the dataset to be converted into dimensionless form; second, these dimensionless representations are normalized using the Euclidean norm to produce quantum-like state vectors; third, the normalized data are mapped into quantum-inspired representations that mimic the structure of quantum states; and fourth, both pure and mixed quantum states are constructed from these representations to encapsulate the statistical and structural properties of the original dataset. This methodology facilitates the emergence of a quantum-inspired representation of classical data, enabling the application of quantum information-theoretic tools for enhanced insight and computational advantage.

Consider a 9-dimensional quantum feature space of bases which describes the weather dataset

Weather variables are obtained in terms of bases such as the maximum temperature , the minimum temperature , the dew point , the cloud cover , the humidity , the precipitation , the pressure , the x-component of the wind speed and the y-component of the wind speed , where the x, y-components of the wind speed are resolved in terms of wind speed and degree of wind direction, respectively.

Thus, an arbitrary instance of a pure quantum weather state expresses the numerical data of the row in the weather dataset and can be written as:

17

where are real values, d is the day order in the month, m is the month order in the year, and y is the year. For example, if the date is 02/01/2009, then the corresponding daily quantum weather state of the same date is written as Consequently, the quantum daily weather state can be described by numerical data in terms of nine bases as follows⁴¹:       

18

The daily quantum weather state has distinct units. Therefore, all data must be reformed into dimensionless data to be weighted. This goal is achieved by calculating the maximum value of each variable for each column from January 1, 2009, to December 31, 2019, in the statistics of weather data. The maximum values of eight variables are calculated to construct the maximal reference and are listed in Table 1.

Table 1.

Maximal weather eight variables reference.

Varaible	Max_Ref	Varaible	Max_Ref
MaxTemp (K)	308	Humidity %	98
MinTemp (K)	296	Precip (mm)	80
DewPoint (K)	292	Pressure (mbar)	1033
Cloud Cover %	100	WindSpeed (km/h)	35

Thus, the daily quantum weather state can be reformed in dimensionless data by applying Table 1, where each value of a weather variable is divided by the corresponding maximal value of the same weather variable in what follows:

19

where Eq. (19) is non-normailzed. Subsequently, the state is normalized by Eq. ( 2) to be⁴¹:

20

where Eq. (20) is the normalized daily quantum weather pure state. The same computations are applied to the overall instance weather states to obtain normalized pure daily quantum weather states. According to the statistics, we have 4017 normalized daily pure quantum weather states from January 1, 2009, to December 31, 2019.

The state is built over an arbitrary date interval from the row to the j-row. An arbitrary date interval may be a week, month, season, or year. In this framework, the date interval is always a month. The first step, the weather dataset is modified into a new formalism where each column is divided by the corresponding maximal values according to Table 1 to convert it into dimensionless data . Thus, is partitioned to 132 new dimensionless weather dataset where and gives a certain month and is presented by the rectangular matrix of In the second step, the state is formed from the row to the row by Eqn.(3) as:

21

where is the month number, is the year number and is the transpose of the matrix and the trace aims to find the normalized state .

Now, the classical data are reformed based on months as the time unit. Thus, the classical data are formulated into 132 non-normalized, monthly dimensionless matrices, each with a different number of rows corresponding to each month. Consequently, the classical information system is partitioned into 132 non-normalized monthly dimensionless rectangular matrices and The following step, the classical information is transformed into the quantuminformation system as the square matrices January 2009, Feb/2009, March 2009, , October 2019, November 2019, and December 2019, respectively, by Eq. (21). The monthly state of January in 2009 is computed from 01/01/2009 to 31/01/2009 by^41,42:

22

The representation matrix of the state is provided by^41,42:

23

It is noted that is symmetric. The pressure occupies the front with the maximum value 0.49052, while the y-wind speed component comes in the last position with a minimal value 0.00024. In addition, the row or column of pressure has the highest values among other weather variables. The matrix contains only negative elements between the cloud cover and the wind speed component with . The minimal value represents the interaction between the precipitation and the wind speed component with 0.00008 in the matrix. Hence, the element value has a physical nature between two weather variables, except for diagonal elements, which express a weather variable. All elements affect the trace value. Hence, the effects of all weather variables appear in each element, diagonal or non-diagonal. Other monthly states and are obtained similarly to obtain representation matrices. The number of monthly states is 132. We cannot write all the matrices. We give only an example for pure and mixed states as daily and monthly quantum weather states, respectively. All daily and monthly states are tested using quantum informative measures, such as fidelity, classical information, quantum information, decoherent information, quantum stability index, Shannon entropy of CI, Shannon entropy of QI, and von Neumann entropy, respectively.

Discussion and results

In the previous section, the classical data is transformed into a quantum-inspired system as a quantum information system relative to the maximum reference in Table 1 from 2009 to 2019. The first type of state is a 4017 daily pure weather state. The second type of state includes 132 mixed-weather monthly states obtained using square matrices. Hence, the corresponding mathematical model is prepared and the quantum information system is created. The quantum information system is ready to be checked using the quantum informative relations. In this section, the prepared quantum information system is investigated in detail. In the following analysis, we compute the fidelity F, classical information CI, quantum information QI and decoherent information DI, Shannon entropy of CI,  Shannon entropy of QI,  von Neumann entropy of DI,  and quantum stability index . The San Francisco weather dataset is presented briefly, accompanied by eight figures for eight variables of the dataset ²¹. The maximum temperature, minimum temperature, dew point, cloud cover, humidity, precipitation, pressure, and wind speed are plotted versus days in Fig.1. All figures take nearly identical waveforms of distinct amplitudes, except for precipitation. The behavior of the weather variables is nearly regular; otherwise, for precipitation.

Fig. 1.

San Francisco weather features. (a) Minimum temperature. (b) Maximum temperature. (c) Dew point. (d) Cloud cover. (e) Humidity. (f) Precipitation. (g) Pressure. (h) Wind speed.

Fidelity

Here, fidelity is used to test the efficiency of the maximal reference in Table 1, by Eq. (5), where is the dialy pure state and is the monthly state. We computed the fidelity for 4017 values. Therefore, not all fidelity values can be tabulated here. Therefore, Table 2 is constituted instead of tabulating fidelity values to represent important values where the factor is applied to all values in the table. Some facts about the years 2009–2019 are displayed.

Table 2.

Fidelity values of monthly states , based on years from 2009 to 2019.

	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
	982	981	981	982	983	984	983	983	984	984	984
	913	923	918	921	917	928	936	927	920	933	901
	610	732	526	645	495	472	681	690	609	671	506
	333	191	381	261	436	442	248	221	202	250	433
	57	45	59	59	59	61	48	55	51	51	67
[0,0.5]	0	0	0	0	1	1	0	0	0	0	0
(0.5,0.6)	0	0	4	0	2	1	0	0	0	0	4
[0.6,0.7)	4	0	3	2	6	4	1	1	1	5	1
[0.7,0.8)	14	11	14	15	31	12	13	15	14	12	23
[0.8,0.9)	139	135	125	122	113	116	95	104	124	107	147
[0.9,1]	208	219	219	227	212	231	256	246	226	241	190

In the table, days are presented horizontally, while some fidelity values and the number of days that describe fidelity intervals are vertically tabulated for a fixed month. is the maximal fidelity value per month, is the average fidelity value per month, is the minimal fidelity value per month, is the fidelity range and is the standard deviation of fidelity. Here, there are six fidelity intervals, and several days are mentioned for the corresponding intervals. Regarding fidelity values, it is eminent that is closer to than despite the large for all years to two intervals [0.8, 0.9) and , respectively. The fidelity values are generally distributed as 2, 11, 28, 174, 1327, and 2475 in [0, 0.5], (0.5, 0.6), [0.6, 0.7), [0.7, 0.8), [0.8, 0.9), and [0.9, 1], respectively. All values of , and are located in [0.9, 1] while the values of are located in [0, 0.5], (0.5, 0.6),[0.6, 0.7),  and [0.7, 0.8). The values do not exceed 0.984 with a minimum value of 0.981. The values are enclosed between 0.901 and 0.936. The values take the minimal value 0.472 and the maximal value 0.732. The values are between 0.045 and 0.06. Thus, the interval is located between [0.8, 0.9), and [0.9, 1]. Therefore, the major weather data is more coherent with respect to the maximal reference. In Fig. 2, the fidelity is plotted versus the months from January 2009 to December 2019. The fidelity is evident in the waveforms of varying amplitudes. The fidelity ranges between 0.472 and 0.984. Thus, it is notable that the fidelity inspects the maximum weather reference of Table 1. The majority of days have high fidelity values that are greater than 0.8. This maximum weather reference is certainly considered a good reference. Hence, this maximal weather reference can be used to measure other quantum informative relations.

Fig. 2.

Fidelity.

Predictions

In this context, quantum informative measurements are non-linear waveforms. They are numerical values and do not have a functional formula. Consequently, we expect the functional formula for each quantum informative measurement. The Fourier series is an adequate functional formula. Therefore, we expect two Fourier series forms for more accuracy and precision. Let us suppose that the Fourier series consists of a free term, ten cosine terms, and ten sine terms, respectively, expressed as:

24

where are real parameters. The Fourier series parameters are later computed using computational values of quantum informative measurements; CI, QI, DI, and are found by Eqs. (7–16). The fitting curve method is used as the technique to find the parameters of the Fourier series as follows:

25

where is the predicted value at time and represents the corresponding computed value for the quantum measurement f.

The parameters of the first Fourier series are determined by numerical computational values of each quantum informative measurement from January 2009 to December 2017, while the parameters of the second Fourier series are deduced by numerical computational values of the same quantum informative measurement from January 2009 to December 2019. The first and second Fourier series depend on 108 values and 132 values, respectively. Later, we must estimate some future values of quantum informative measurements using two Fourier series. Therefore, the Fourier series can be utilized to predict future month values of the years 2020 and 2021, respectively. The parameters of the first and second Fourier series are tabulated in Tables 3 and 4, respectively, for , , , , and . Two series are checked comparing with the computational values of quantum informative measurement in three absolute errors; and , where q is the quantum informative measurement. Also, the Pearson correlation of the three absolute errors is calculated to distinguish between the set of computational values and the two Fourier series to recognize the quality of future predictions.

Table 3.

Fourier coefficients of .

	0.256727	0.665949	0.077324	0.714385	3.87327	0.092477	15.4905
		0.004613	0.001830	0.006744	0.069861	0.000902	0.103322
	0.023401		0.036820			0.038794	4.25743
		0.008495	0.001130	0.010414	0.089924	0.000607	0.169416
	0.012558		0.006489			0.007563	0.724654
		0.006174	0.001808	0.009397	0.064774	0.001001	0.202049
	0.008151		0.002083			0.002196
		0.007259		0.003441	0.001578
			0.001956	0.000965	0.008258	0.002055	0.255693
		0.0102142		0.008456	0.103631
	0.000233		0.001292	0.000309	0.004666	0.000326	0.054006
	0.000483		0.003789			0.003674	0.468069
	0.025125		0.030581			0.031946	3.72356
	0.006883					0.000037
	0.004139	0.009241
	0.004036		0.001678			0.0012615	0.126739
	0.005680						0.060856
	0.004031		0.000018			0.001087
	0.001476		0.002778			0.001937	0.343117
	0.001481
	0.000005		0.001031	0.000795	0.025382		0.097956

Table 4.

Fourier coefficients of .

	0.253789	0.66822	0.077991	0.717926	3.89144	0.093272	15.5578
		0.004456	0.001714	0.006971	0.082208	0.000199	0.064082
	0.022665		0.037811			0.039989	4.34915
		0.005599	0.000805	0.006489	0.053243	0.000447	0.143078
	0.010351		0.004336			0.004906	0.494278
		0.002348	0.002974	0.006430	0.016627	0.002995	0.355583
	0.005273
		0.004698		0.000498
		0.000309	0.001079	0.001348	0.014178	0.001397	0.191005
		0.009391		0.006068	0.078732
	0.000670		0.000280		0.006696
			0.006548	0.001924		0.006784	0.771477
	0.023752		0.029838		0.002222	0.030745	3.62711
	0.005332		0.002353			0.002849	0.178985
	0.003851	0.008981
	0.003855		0.001797			0.001751	0.135848
	0.004042
	0.003791
	0.000562		0.000149			0.000330	0.051087
	0.001467	0.000254
		0.000349	0.000333	0.001639	0.035974	0.000039	0.063612

Classical information

The classical information values of the monthly states are calculated for consecutive months by Eqs. (7, 21, 22, 23) and are listed in Table 5. From 2009 to 2019, has a minimal value of 0.20261 in August 2015 and a maximum value of 0.40266 in December 2013. Consequently, ranges between 0.20261 and 0.40266. generally declines from 2009 to 2019 for all months. From May to October, is characterized by small values and variations, but in other months, it has large values and variations, especially in January and December. Therefore, the variation of takes nonlinearly wavily forms of different amplitudes from the maximal value in January to the minimal value in July and August and reaches another maximal value in December every year.

Table 5.

Classical information .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	31904	26371	31749	32850	37278	33714	32916	24316	25949	25298	26100
Feb	26508	27192	30993	30948	36357	26027	24513	26263	22585	31362	25267
Mar	28413	29308	26067	25691	28854	24861	26047	22323	25278	25990	24211
Apr	29487	28267	27807	26261	25955	26580	27387	24416	24214	25598	21731
May	24638	28171	27464	27779	25873	25025	24076	23391	24560	24049	21860
Jun	23943	24784	23669	24740	22495	24225	22550	23039	22345	23293	21923
Jul	24925	25424	23353	23307	23440	21729	20632	22176	22638	21846	22437
Aug	23599	24787	24202	23369	22593	22062	20261	22139	20836	21395	21124
Sep	22748	22947	23177	24585	22184	21656	21383	22470	20729	22447	21083
Oct	23795	22748	22660	22974	26546	22193	20875	20849	24443	22314	25649
Nov	28764	27096	29657	24184	28090	23939	28580	23596	23317	25070	24993
Dec	30958	25556	38395	25752	40266	22207	28433	28324	33156	26972	22890

In Table 6, and are good correlations, while has an excellent correlation. The second correlation provides slightly better results compared to the first prediction. In Table 7, at Sep/2020 and at January 2020. at Sep/2021 and at Jul/2021. The years 2020 and 2021 behave the same as in previous years. It is evident that and have irregular wavily behaviors and vary annular in Fig. 3. Three curves are near for most of the points.

Table 6.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of CI.

	0.00017	0.00036	0.00021
	0.01843	0.01816	0.00436
	0.10665	0.08750	0.01914
	0.01832	0.01759	0.00415
P	0.72818	0.73487	0.99088

Table 7.

Predictions of , for 2020 and 2021.

Y	2020			2021
M
Jan	30173	29370	803	31169	30251	918
Feb	26809	27548	739	28932	28394	538
Mar	26224	26339	115	26288	25877	411
Apr	25875	25675	199	27478	26570	908
May	26389	26040	349	24983	24433	550
Jun	24135	23877	258	23152	22952	201
Jul	22978	22808	170	23040	22963	77
Aug	23506	23062	444	22068	21858	209
Sep	22287	22308	21	22444	22296	148
Oct	22991	22896	95	23126	23443	317
Nov	23781	23975	194	28318	27887	431
Dec	26447	26632	185	33555	31640	1914

Fig. 3.

Classical information CI.

Quantum information

Quantum information QI is checked for monthly states such as CI and of monthly states, which are also computed for each month by Eqs. (8, 21, 22, 23) and are listed in Table 8. From 2009 to 2019, ranges between the minimal value of 0.45402 in Dec. 2013 and the maximum value of 0.77238 in Aug. 2015. Clearly, reaches a maximum in July and August and decreases minimally in January and December based on the annular. Therefore, the variation of oscillates in nonlinearly sinusoidal forms of different amplitudes from the minimal value in January to the maximal value in July and August and returns to another minimum value in December every year. In regarding of behaves inversely behavior of If increases, then decreases and vice versa. Also, by comparing Tables 5 and 8, is always greater than for all months and years. Thus, the effect of is stronger than the effect of in this model. Without repetitions, other tables are designed completely as CI tables. In Table 9, the first and the second predictions of and give excellent results especially and Both of and are greater than and respectively. In Table 10, the lowest absolute errors are in March 2020 and in August 2021. The greatest absolute errors are at January 2020 and at Dec/2021. and have the same behaviors in 2020 and 2021 as in past years. It is obvious that and change irregularly in Fig. 4. In particular, and occupy opposite positions of and .

Table 8.

Qunatum information .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	54761	62875	56698	55526	49700	54131	58854	67203	60703	66339	60983
Feb	62135	67068	52886	58247	57371	62480	64853	63624	64828	56760	57850
Mar	57552	60676	64717	61189	62733	66299	69104	65727	64011	63894	62611
Apr	59233	61187	62691	61690	60781	66709	65558	68385	63918	68220	68447
May	65438	61516	63887	67813	65988	68575	72168	71212	68769	71039	70316
Jun	71467	70361	68916	70044	69867	73178	73863	73191	72965	73300	72107
Jul	73285	73147	73885	74160	74818	75831	76719	75597	75547	76605	75799
Aug	73399	72413	74806	74652	75749	76754	77238	76808	76528	75840	75714
Sep	73646	71155	74011	73756	73908	76010	74557	74039	72960	75202	70233
Oct	64159	67277	69617	70290	63073	71154	74766	70622	68247	72550	61098
Nov	62083	61354	61086	65019	60209	64280	60011	68570	69591	60924	61913
Dec	56893	65195	47585	60366	45402	65341	58297	55378	56068	63758	66588

Table 9.

Min, Avg, and Max of absolute Errors, standard Deviation, and Pearson Correlation of QI.

	0.00003	0.00039	0.00008
	0.02650	0.025470	0.00592
	0.12861	0.10705	0.02379
	0.02227	0.02205	0.00532
P	0.86751	0.87341	0.99324

Table 10.

Predictions of .

Y	2020			2021
M
Jan	58936	60137	1200	56456	57382	926
Feb	63704	62575	1129	60220	59670	550
Mar	62809	62800	8	63669	63619	51
Apr	64947	65779	832	62558	63454	895
May	67045	67724	678	66952	67551	599
Jun	71712	72087	375	71886	71937	51
Jul	74861	75217	356	74223	74418	195
Aug	74816	74952	136	76302	76323	21
Sep	74200	74523	323	72963	72344	619
Oct	69308	69791	483	68880	67783	1097
Nov	65344	64655	689	61680	61494	186
Dec	61079	61406	327	53728	56107	2379

Fig. 4.

Quantum information QI.

Decoherent information

In this subsection, the decoherent information DI is discussed in addition to the aforementioned CI and QI. In a similar way, values of monthly states are also computed by Eqs. (10, 21, 22, 23) for all months and are listed in Table 11. A range of lies between the minimal value of 0.00992 in August 2011 and the maximum value of 0.16884 in February 2019 from 2009 to 2019. There is a near similarity between and , where for the same month. Similarly, other tables are formed with the same techniques of CI and QI.

Table 11.

Decoherent information .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	13336	10755	11553	11624	13022	12155	8230	8481	13348	8363	12917
Feb	11357	5740	16120	10805	6272	11494	10634	10113	12587	11879	16884
Mar	14035	10016	9215	13120	8413	8840	4849	11951	10712	10116	13178
Apr	11279	10546	9502	12049	13264	6711	7055	7198	11868	6182	9822
May	9924	10313	8649	4408	8139	6399	3756	5396	6671	4912	7824
Jun	4590	4855	7415	5215	7638	2598	3587	3771	4689	3407	5970
Jul	1790	1429	2762	2533	1742	2440	2648	2227	1815	1549	1764
Aug	3002	2800	992	1978	1658	1184	2501	1053	2636	2765	3162
Sep	3606	5898	2812	1659	3908	2334	4060	3491	6311	2351	8683
Oct	12045	9976	7723	6736	10381	6653	4360	8530	7309	5135	13253
Nov	9153	11550	9257	10797	11701	11781	11409	7834	7091	14006	13094
Dec	12148	9249	14021	13881	14332	12452	13269	16298	10776	9270	10521

In Table 12, the second prediction gives more accurate results that are distinguishable from the first prediction with less absolute errors. Also, and are excellent correlation results such that with a slight difference since . and are very close to and respectively. Hence, the predictions of QI and DI are better than the predictions of CI. In Table 13, the lowest absolute errors are in March 2020 and in January 2021. The greatest absolute errors are in December 2020 and in February 2021. and move in the same direction in 2020 and 2021 as in previous years. Eminently and curves behave geometrically and respectively but differ in computational ranges in Fig. 5.

Table 12.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of DI.

	0.00022	0.00008	0.00008
	0.01603	0.01575	0.00373
	0.06036	0.05664	0.01088
	0.01309	0.05664	0.00276
P	0.86597	0.87296	0.99200

Table 13.

Predictions of , for 2020 and 2021.

Y	2020			2021
M
Jan	10891	10493	398	12375	12368	8
Feb	9487	9876	389	10847	11936	1088
Mar	10968	10861	107	10043	10505	461
Apr	9179	8546	633	9964	9977	13
May	6566	6236	330	8066	8016	49
Jun	4153	4036	117	4961	5111	150
Jul	2161	1975	185	2736	2619	118
Aug	1679	1986	308	1631	1819	188
Sep	3512	3168	344	4593	5360	767
Oct	7701	7314	387	7994	8774	780
Nov	10875	11370	495	10002	10619	617
Dec	12475	11963	512	12718	12253	465

Fig. 5.

Decoherent information DI.

Shannon entropy of CI

The purpose of Shannon entropy is to measure the randomness of CI. Therefore, the Shannon entropy values are determined for monthly states by Eqs. (14, 21, 22, 23) for each month and are recorded in Table 14. From 2009 to 2019, has the lowest randomness value of CI with 0.57329 in December 2013, and the highest randomness value appears in October 2016 with 0.78533. Hence, here takes a range between 0.57329 and 0.78533.

Table 14.

Shannon entropy of CI .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	65106	71822	63989	64264	59271	63376	63600	73996	72258	72682	72428
Feb	71641	70007	66337	66447	58517	72345	74566	72189	76734	65747	74845
Mar	69567	68131	71570	72784	68421	73635	71423	77366	73067	72608	75399
Apr	67865	69687	69274	71949	72390	70828	70272	73635	74837	71823	77141
May	73163	69564	70096	68273	71388	71717	73896	74225	72573	73170	77372
Jun	72408	71305	73966	71393	74837	71225	74549	73536	74274	72538	74835
Jul	69453	68873	71933	72253	71615	74274	76684	74812	72493	73613	73172
Aug	71180	69844	70048	71651	72717	73230	76988	74513	75651	75193	74912
Sep	72258	72963	71814	70359	73544	74252	75496	73389	75735	73212	75471
Oct	74474	75343	74850	74099	70582	74767	76237	78533	71844	74169	70769
Nov	68660	70323	67885	74934	70093	74365	69052	74527	75101	73289	73177
Dec	65620	71805	58377	73212	57329	77438	69875	69772	65300	71104	76103

For the same month, often varies with small changes from year to year. October occupies the first month of randomness of CI and is followed by June. Clearly, oscillates in the same year. It is noted that increases starting in 2016. Generally, the randomness of CI is considered high. In Table 15, the minimum, average and maximum absolute errors are suitable, as well as the standard deviations. For Pearson correlation, the results are not good where and Consequently, the predictions are not strong and are mean. In Table 16, two predictions and are mentioned for 2020 and 2021. at Sep/2020 and at Feb/2020. at Jul/2021 and at Dec/2021. In Fig.6, and take nonlinear waveforms but the curve grows up starting from 2016. The randomness of CI gradually increases in the nonlinear waveform from 2016 to 2019.

Table 15.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of .

	0.00069	0.00003	0.00050
	0.02349	0.02330	0.00541
	0.12058	0.10982	0.02186
	0.02209	0.02103	0.00501
P	0.57369	0.58279	0.98439

Table 16.

Predictions of CI , for 2020 and 2021.

Y	2020			2021
M
Jan	67403	68241	838	65927	67087	1160
Feb	71160	70290	870	68516	69498	982
Mar	72189	72076	114	71754	72421	667
Apr	72123	72236	113	70136	71264	1128
May	70591	70966	375	72818	73590	772
Jun	71936	72160	224	73638	73855	217
Jul	72782	72885	102	72564	72614	50
Aug	71782	72488	706	73427	73759	332
Sep	73544	73491	53	73439	73662	223
Oct	74648	74506	142	74115	73700	415
Nov	74752	74530	222	69499	69949	450
Dec	71735	71523	213	64045	66231	2186

Fig. 6.

Shannon entropy .

Shannon entropy of QI

In the same way, the Shannon entropy is used to inspect QI similar to CI in the previous subsection. Shannon entropy values of monthly states are calculated for each month by Eqs. (15, 21, 22, 23) and are listed in Table 17.

Table 17.

Shannon entropy of QI.

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	3.218	4.203	3.244	3.556	3	3.065	3.326	4.272	4.188	4.001	3.973
Feb	4.158	4.055	3.644	3.449	2.950	4.176	4.116	3.761	4.483	3.167	4.663
Mar	3.591	3.706	4.214	4.304	3.730	4.178	4.169	4.477	3.881	4.205	4.398
Apr	3.456	3.918	3.619	3.952	3.649	4.009	3.921	4.125	4.027	4.003	4.061
May	3.933	3.626	3.772	3.534	3.743	3.764	4.470	4.150	3.840	4.125	4.545
Jun	3.980	3.709	4.160	3.729	3.951	3.883	4.290	4.067	4.032	3.882	4.041
Jul	3.561	3.520	3.862	3.927	3.875	4.083	4.434	4.311	3.929	3.988	4.036
Aug	3.618	3.697	4.076	3.739	3.982	3.968	4.423	4.334	4.229	4.190	4.045
Sep	3.539	3.899	3.693	3.732	3.756	4.020	4.226	3.963	3.970	3.932	3.705
Oct	3.770	4.153	4.138	4.095	3.580	3.967	4.146	4.450	3.527	3.865	3.116
Nov	3.602	3.766	3.691	4.190	3.622	3.960	3.476	4.076	4.198	3.880	3.765
Dec	3.555	3.634	2.993	4.243	2.590	4.406	3.901	3.694	3.209	3.681	4.416

Obviously, all terms of are greater than one since some nondiagonal terms are 72 terms greater than the number of dimensions of the quantum feature space. From 2009 to 2019, a range of is located between the minimal value of 2.590 in December 2013 and the maximal value of 4.663 in February 2019. In fact, is similar to with different ranges. In Table 18, the absolute errors are close to the standard deviations. Two Pearson correlations are weak where and Also, is less than It is prominent that predictions of randomness and are not accurate. In Table 19, two predictions and are recorded for 2020 and 2021. at Sep/2020 and at Dec/2020. at Jun/2021 and at January 2021. In Fig.7, and curves are similar to curves geometrically with different ranges. Consequently, curves of and behave physically.

Table 18.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of .

	0.00107	0.00476	0.00083
	0.26118	0.24941	0.11325
	1.13172	0.97461	0.29708
	0.23359	0.20060	0.08633
P	0.40453	0.46559	0.86890

Table 19.

Predictions of , for 2020 and 2021.

Y	2020			2021
M
Jan	3.579	3.718	0.139	3.278	3.575	0.297
Feb	3.915	3.816	0.099	3.641	3.925	0.283
Mar	4.053	4.094	0.041	3.993	4.063	0.070
Apr	3.954	4.062	0.108	3.606	3.739	0.133
May	3.664	3.803	0.138	4.053	4.101	0.048
Jun	3.732	3.865	0.133	4.065	4.066	0.001
Jul	3.904	3.981	0.078	3.923	3.920	0.004
Aug	3.724	3.944	0.221	3.913	4.024	0.111
Sep	3.933	3.948	0.015	3.794	3.761	0.033
Oct	3.971	4.025	0.054	3.926	3.782	0.144
Nov	4.113	4.088	0.025	3.575	3.646	0.070
Dec	4.130	3.919	0.211	3.284	3.529	0.245

Fig. 7.

Shannon entropy based on months.

Von Neumann Entropy of DI

von Neumann entropy deals with the decoherent information DI to measure its randomness. values of the monthly states are computed for each month by Eqs. (16, 21, 22, 23) and are listed in Table 20. As shown in the table, varies from the lowest value of 0.01717 in August 2011 to the highest value of 0.17635 in February 2011. January and December have the highest von Neumann entropies, while July has the lowest von Neumann entropy. In Table 21, the absolute errors and standard deviations are very small relative to other measurements of quantum information. In addition to the strong correlation between the computational data and the predicted data and . The Pearson correlation is strong with values such as and Therefore, two predictions are more accurate in Table 22. in March 2020 and in April 2020. in May 2021, and in October 2021. In Fig.8, and curves are near to CI curves computationally and geometrically. Hence, and CI coincide in physical characteristics.

Table 20.

Von Neumann entropy of DI .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	14933	12059	13264	13203	14979	13412	10085	10279	14305	10155	14283
Feb	12381	7271	17653	12920	8408	12915	12397	12145	13162	13316	18195
Mar	15307	11865	10593	14602	10269	10501	6480	12614	12284	12106	14457
Apr	13571	12295	11243	13744	14809	8418	8933	9020	13090	7857	11277
May	11202	11429	10288	6000	10012	8363	4989	7030	8323	6154	9326
Jun	5958	6768	8909	7072	9114	3937	4838	5233	6092	4832	7819
Jul	2849	2331	3942	3714	2775	6573	3689	3258	2765	2454	2722
Aug	4455	4035	1717	3048	2661	2002	3650	1755	3713	3754	4386
Sep	5233	7424	4149	2671	5536	3509	5489	4905	8013	3439	10490
Oct	13745	11312	9548	8621	12320	8443	5889	10020	9415	6914	15767
Nov	11449	13609	11608	12570	13659	13393	13580	9771	8840	15101	15073
Dec	14231	10991	15840	15618	16622	13892	15035	16978	12902	11023	12833

Table 21.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of .

	0.00022	0.00008	0.00006
	0.01586	0.01568	0.00387
	0.06598	0.05504	0.01481
	0.01314	0.01225	0.00367
P	0.87848	0.88684	0.99057

Table 22.

Predictions of for 2020 and 2021.

Y	2020			2021
M
Jan	12417	12140	277	13946	13924	23
Feb	11208	11481	273	12309	13346	1037
Mar	12445	12491	46	11523	11985	462
Apr	10855	10181	674	11765	11678	87
May	8209	7830	379	9536	9542	6
Jun	5736	5564	172	6420	6578	158
Jul	3293	3078	215	3730	3675	54
Aug	2600	2909	310	2902	2891	11
Sep	4812	4435	377	6008	6995	987
Oct	9328	8962	366	9169	10650	1481
Nov	12699	13059	360	12357	12773	417
Dec	13914	13448	466	14768	14240	529

Fig. 8.

Von Neumann entropy .

Quantum Stability Index

The quantum stability index (QSI), denoted by , is introduced mathematically in Sect. "Quantum model of dataset" as a novel quantum informative metric. QSI is a new formula that is defined here to assess the stability of the quantum information system. QSI is designed in terms of three distinct information types; CI, QI, and DI. QSI quantifies the stability by analyzing the relationship among CI, QI, and DI as previously discussed. QSI generally discusses four physical situations for the stability of the quantum information system. The first situation describes the full stable quantum information system when . The second situation expresses the stable quantum information system when . In the third situation, the quantum information system becomes ambiguous for . No one can say that the system is metastable. When , the fourth situation illustrates that the quantum information system begins to lose stability when travels away . The values of the quantum stability index of the monthly states are evaluated in degrees for individual months by Eqs. (13, 21, 22, 23), and are listed in Table 23. For a lot of details about the quantum stability index, it has been inspected for months.

Table 23.

Quantum stability index .

M/Y	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019
Jan	21.42	19.14	19.87	19.93	21.15	20.40	16.67	16.93	21.43	16.81	21.06
Feb	19.69	13.86	23.67	19.19	14.50	19.82	19.03	18.54	20.78	20.16	24.26
Mar	22	18.45	17.67	21.24	16.86	17.30	12.72	20.22	19.10	18.55	21.29
Apr	19.62	18.95	17.95	20.31	21.36	15.01	15.40	15.56	20.15	14.4	18.26
May	18.36	18.73	17.10	12.12	16.58	14.65	11.18	13.43	14.97	12.8	16.24
Jun	12.37	12.73	15.80	13.20	16.04	9.28	10.92	11.20	12.51	10.64	14.14
Jul	7.69	6.87	9.57	9.16	7.58	8.99	9.37	8.58	7.74	7.15	7.63
Aug	9.98	9.63	5.72	8.09	7.40	6.25	9.10	5.89	9.34	9.57	10.24
Sep	10.95	14.06	9.65	7.40	11.40	8.79	11.62	10.77	14.55	8.82	17.14
Oct	20.31	18.41	16.14	15.04	18.80	14.95	12.05	16.98	15.69	13.10	21.35
Nov	17.61	19.87	17.71	19.18	20	20.07	19.74	16.25	15.44	21.98	21.21
Dec	20.40	17.71	21.99	21.87	22.25	20.66	21.36	23.81	19.16	17.73	18.93

The quantum stability index changes from minimal to maximal limits. For example, it increased from 5.72 in August 2011 to 24.26 in February 2019. For the same month, appears in oscillating forms and varies yearly. The months: January, February, November, and December have the highest variance. In contrast, July and August have less variance. Years change in irregular sinusoidal forms. All terms of are less than 45 with a difference of 20.74 at least and extending to 39.28. In Table 24, the average absolute errors approach the minimal absolute errors while diverging for the maximal absolute errors. Standard deviations and average absolute errors form suitable intervals of the stability as and for and respectively. Pearson correlations are very strong where and In Table 25, in September 2020 and in April 2020. in May 2021 and in September 2021. In Fig. 9, and curves look like all quantum informative measurements in nonlinear waveforms. Three curves are very close to the major points.

Table 24.

Min, Avg, and Max of absolute errors, standard deviation, and Pearson correlation of .

	0.034	0.002	0.072
	1.79	1.72	0.54
	5.77	6.04	1.65
	1.34	1.25	0.43
P	0.88552	0.89691	0.98730

Table 25.

Predictions of , for 2020 and 2021.

Y	2020			2021
M
Jan	20.03	18.85	1.18	20.14	20.63	0.49
Feb	17.52	18.20	0.68	18.8	19.9	1.10
Mar	19.02	19.20	0.18	17.81	18.76	0.96
Apr	18.13	16.85	1.27	18.05	18.26	0.21
May	15.1	14.31	0.79	16.37	16.29	0.07
Jun	11.81	11.45	0.36	13.22	13.07	0.15
Jul	8.28	8.12	0.16	9.1	8.81	0.29
Aug	7.99	7.87	0.12	7.53	8.12	0.58
Sep	10.15	10.04	0.11	11.37	13.01	1.65
Oct	15.98	15.54	0.43	16.43	16.99	0.56
Nov	19.97	19.71	0.26	19.71	18.95	0.14
Dec	19.08	20.02	0.31	21.27	20.44	0.83

Fig. 9.

Quantum stability index in degree .

Results and analysis

We provide a brief overview of the San Francisco weather model, which is processed using quantum information theory. The classical information system is transformed into a quantum-inspired system using the maximal reference and Euclidean norm to form daily pure and monthly mixed states. The pure daily states of each month are treated with the same monthly state in fidelity to measure the quality of the weather. Most days give excellent results with high fidelity greater than 0.8, especially in July and August from 2009 to 2019. Compared to January and December in the same interval, their fidelity values are smaller. According to the strength effect, three types of information are organized: quantum information, classical information, and decoherent information. Two predictions of QI and DI are close to their computational values because Pearson correlations are excellent. For CI, Pearson correlations are good and the two corresponding predictions are good results. Every type of entropy takes the same arrangement as the type of information. The Shannon entropy of CI is given by nine terms expressing diagonal elements of the state and is always less than one. The Shannon entropy of QI is described by 72 terms that express non-diagonal elements of the state and are always greater than one. Two predictions of the Shannon entropy of CI and QI are moderate because the Pearson correlations are moderate and weak. For von Neumann entropy, the Pearson correlations are strong, and the two predictions yield excellent results. In the end, the quantum stability index measures the stability of the quantum information system. It confirms that the instability limit is so far. In contrast. It is more stable. Hence, the weather in San Francisco is moderate from 2009 to 2019 with computational data and continues to moderate from 2020 to 2021 using predictions.

Conclusion

We employ quantum information theory as a novel analytical lens for examining weather data, aiming to derive meaningful insights that may open new avenues for understanding climate dynamics. In this study, the San Francisco weather dataset, denoted by , is modeled using a quantum-inspired framework. This framework enables the application of quantum metrics to identify complex patterns within the data. To ensure transparency and methodological rigor, we acknowledge the limitations of quantum-inspired representations. Our objective is not to imply that the data possess inherent quantum properties but rather to contribute to the growing interdisciplinary field that applies quantum formalism to classical domains. The forecast component of our work is based on projecting quantum information-theoretic metrics—such as fidelity and stability index—onto the climate data, rather than relying solely on traditional meteorological variables such as temperature or precipitation. These metrics are fitted using Fourier series to uncover temporal patterns and structural characteristics within the dataset. Through this approach, we aim to investigate how these abstract metrics evolve over time and whether they can reveal hidden patterns or anomalies in climate behavior. Ultimately, our goal is to establish a theoretical foundation and demonstrate the feasibility of quantum-inspired representations in climate analysis. In the long term, insights derived from this framework may support informed decision-making by policymakers, contributing to effective climate action and improving environmental conditions. In this regard, quantum information analysis will support understanding and perception of the diverse and multiple mechanisms of climate action and its influence on the environment. At a later stage, based on the inferred results and discovered facts, governments will be able to make effective decisions that limit the negative impacts and support gradual climate reform and the improvement of environmental conditions.

Author contributions

A. Youssef conceptualized and designed the study. D. Nassr, A. Youssef, and Y. Kotb analyzed the data. A. Youssef contributed to the interpretation of the results. A. Youssef and D. Nassr drafted the manuscript. Y. Kotb and H. Bahig reviewed and edited the manuscript. H. Bahig supervised the project. H. Bahig funded the acquisition. All authors approved the final version for publication.

Funding

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammed Ibn Saud Islamic University (IMSIU) (Grant number IMSIU-DDRSP2502).

Data availability

The dataset used in the current study is publicly available at https://www.kaggle.com/datasets/luisvivas/weather-north-america.

Declarations

Competing interests

The authors declare no competing interests.