Quantum entanglement and covariance

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Quantum Entanglement

Quantum entanglement is a phenomenon in which two or more particles become strongly correlated, such that no matter how far apart they are in space, their states remain instantaneously connected.

To understand this more intuitively, consider a hypothetical case involving two quantum coins (A and B) that are entangled. The phenomenon of quantum entanglement can be described as follows:

• Entangled State: Coins A and B are prepared in a state such that their outcomes are always opposite.

• Moment of Observation: If coin A is observed on Earth and lands “heads,” then coin B, even if observed on the Moon, will certainly show “tails.”

• Superposition: However, until the moment of observation, both coins remain in a superposition of “heads” and “tails,” and their states are only determined upon measurement.

At first glance, this phenomenon appears as if information is transmitted faster than the speed of light, seemingly violating the principle of Einstein’s theory of relativity, which states that information cannot travel faster than the speed of light. This apparent contradiction was raised in 1935 by Einstein, Podolsky, and Rosen, in what is now known as the EPR Paradox.

In response to this, physicist John Bell proposed Bell’s Inequality in 1964, which set theoretical limits on the correlations predicted by any local hidden variable theory. Bell’s inequality is expressed as follows:

$$\mathrm{\mid }E(a,b)+E(a,b^{\mathrm{\prime }})+E(a^{\mathrm{\prime }},b)-E(a^{\mathrm{\prime }},b^{\mathrm{\prime }})\mathrm{\mid }\le 2$$

This inequality establishes the maximum correlation possible under classical physics. However, in the 1980s, Alain Aspect and his colleagues conducted experiments using the polarization of photons. These experiments observed results that significantly exceeded the classical limit—reaching approximately 2.828, thereby confirming the predictions of quantum mechanics and demonstrating that quantum entanglement is a real and non-classical phenomenon.

The Difference Between Causality and Probabilistic Correlation in Information Transmission

The correlations observed in quantum entanglement, as demonstrated by Alain Aspect and his colleagues, are determined at the moment of measurement and are inherently probabilistic. The outcomes are not fixed until they are observed. Therefore, no actual information is transmitted between the particles; rather, only statistical correlations are observed simultaneously. This supports the conclusion that the EPR paradox, which assumes a causal relationship, does not apply in this context.

Information transmission can be described as a causal process in which the sending of a signal (cause) leads to its reception (effect). This principle underlies classical communication methods, such as radio waves or fiber optics, where signals travel through a physical medium. In these cases, the temporal order between sending and receiving is well-defined, and the signal’s propagation must occur at or below the speed of light. This respects the principle of causality, meaning that the cause precedes the effect in a one-way flow of information.

In contrast, quantum entanglement is a phenomenon where two or more quantum particles exist in a correlated state. This correlation does not imply a causal relationship. Instead, what is observed between entangled particles is a correlation of measurement outcomes, not a transmission of information. The interaction appears to be bidirectional in nature, but it does not constitute causal information flow.

The key difference is that the correlations observed in quantum entanglement simply indicate a statistical relationshipbetween two events. They do not imply a cause-and-effect connection. In other words, entanglement reflects correlation without causation.

Quantum Entanglement and Wave Function Collapse

Quantum entanglement is explained based on the fundamental nature of the wave function, which represents the state of a particle in quantum mechanics.

The wave function describes the probability distribution of a particle’s properties, such as its position and momentum. The square of the absolute value of the wave function gives the probability density that the particle exists at a specific location.

In this probabilistic framework, a particle exists in a superposition of multiple states until it is observed. It is considered to potentially exist in several different places simultaneously. Upon observation, however, the wave function collapses into a single definite state. This phenomenon is known as wave function collapse.

Quantum entanglement is described by a single, joint wave function that encompasses two or more particles which are mutually connected. This combined wave function includes nonlocal correlations, meaning that once the state of particle A is determined, the state of particle B is instantly determined as well—even if the particles are separated by large distances.

Covariance and Correlation of Random Variables

In statistical analysis, covariance is used to quantify the relationship between two random variables—such as in the case of quantum entanglement. Covariance measures how much two variables change together and is defined as:

$$\mathrm{C}\mathrm{o}\mathrm{v}(X,Y)=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]$$

or equivalently,

$$\mathrm{C}\mathrm{o}\mathrm{v}(X,Y)=\mathbb{E}[XY]-\mathbb{E}[X]\mathbb{E}[Y]$$

Where:

• $\mathbb{E}[X]$is the expected value (mean) of $X$,

• $\mathbb{E}[Y]$ is the expected value of $Y$,

• $\mathbb{E}[XY]$ is the joint expected value of $X$ and $Y$.

There are three main patterns of covariance:

1. Positive Covariance ($\mathrm{Cov}(X,Y)>0$):
   The two variables tend to increase or decrease together. For example, temperature and ice cream sales—when the temperature rises, ice cream sales tend to increase.

2. Negative Covariance ($\mathrm{Cov}(X,Y)<0$):
   The two variables tend to move in opposite directions. For instance, rainfall and the number of outdoor event participants—as rainfall increases, the number of participants tends to decrease.

3. Zero Covariance ($\mathrm{Cov}(X,Y)=0$):
   If the two variables are statistically independent, their covariance is zero. However, zero covariance does not necessarily imply statistical independence. Even if variables are dependent, the overall correlation can cancel out, resulting in zero covariance.

Example: Calculating Covariance

Consider the dataset:
$X=[1,2,3,4,5]$,
$Y=[2,4,6,8,10]$

The means are:

$$\mathbb{E}[X]=3,\mathbb{E}[Y]=\frac{2+4+6+8+10}{5}=6$$

The covariance is calculated as:

$$\mathrm{C}\mathrm{o}\mathrm{v}(X,Y)=\frac{1}{5}\sum _{i=1}^5(X_i-3)(Y_i-6)$$

Breaking it down:

$$\begin{gathered} \mathrm{C}\mathrm{o}\mathrm{v}(X,Y)=\frac{1}{5}[(1-3)(2-6)+(2-3)(4-6)+(3-3)(6-6)+(4-3)(8-6)+(5-3)(10-6)] \\ =\frac{1}{5}[8+2+0+2+8]=\frac{20}{5}=4 \end{gathered}$$

So, the covariance is 4, indicating a positive relationship between the variables—they increase together.

The correlation coefficient, which is a normalized form of covariance, is given by:

$$\mathrm{C}\mathrm{o}\mathrm{r}\mathrm{r}(X,Y)=\frac{\mathrm{C}\mathrm{o}\mathrm{v}(X,Y)}{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}(X)\cdot \mathrm{V}\mathrm{a}\mathrm{r}(Y)}}$$

In the context of quantum entanglement, the relationships between observables of entangled particles can also be expressed in terms of covariance matrices, which represent how measurement outcomes are statistically correlated across different quantum systems.

Applications

1. Portfolio Theory (Financial Engineering)

Background:
In portfolio risk management, the covariance between different assets plays a crucial role in diversifying risk. The lower (or more negative) the covariance between assets, the more effectively the portfolio risk can be reduced.

Example:
Consider the returns of Stock A and Stock B over five periods:

• Returns of Stock A: 5%, 7%, 3%, 6%, 4%

• Returns of Stock B: 2%, 3%, 1%, 4%, 2%

The covariance between these two assets can be calculated using the formula:

$$\mathrm{C}\mathrm{o}\mathrm{v}(A,B)=\frac{1}{N}\sum _{i=1}^N(A_i-\bar{A})(B_i-\bar{B})$$

Where:

• Ai​ and Bi​ are the returns of Stock A and Stock B at timei,
• Aˉ and Bˉ are the mean returns of Stock A and B, respectively,

• $N$ is the number of observations.

This calculation helps investors assess how the returns of two assets move together, enabling more informed decisions on portfolio composition and risk mitigation.

import numpy as np

A = [5, 7, 3, 6, 4]
B = [2, 3, 1, 4, 2]

cov_matrix = np.cov(A, B, bias=True)
print(cov_matrix)

2. Principal Component Analysis (PCA)

Background:
PCA is a technique used to compress high-dimensional data into a smaller number of principal axes (principal components) for purposes such as visualization and classification. It reduces dimensionality by computing the eigenvectors and eigenvalues of the covariance matrix of the data.

Example:
PCA is commonly applied to dimensionality reduction of image data or audio data, enabling efficient storage, noise reduction, and pattern recognition.

from sklearn.decomposition import PCA
import numpy as np

# sample data
data = np.array([[2.5, 2.4], [0.5, 0.7], [2.2, 2.9], [1.9, 2.2], [3.1, 3.0], [2.3, 2.7], [2.0, 1.6], [1.0, 1.1], [1.5, 1.6], [1.1, 0.9]])

# PCA execution
pca = PCA(n_components=2)
pca.fit(data)

print("covariance matrix:")
print(np.cov(data.T))
print("n principal component:")
print(pca.components_)

3. Signal Processing (Audio and Image)

Background:
Covariance matrices are used to capture correlations between signals, especially for tasks such as noise reduction and data compression.

Example:
Applications include filtering of audio signals and noise reduction in images, where identifying and removing redundant or noisy components improves signal quality and efficiency.

import numpy as np
from scipy.io import wavfile

# Loading voice data
rate, data = wavfile.read("sample.wav")

# Calculation of the covariance matrix
cov_matrix = np.cov(data.T)
print(cov_matrix)

4. Recommendation Systems

Background:
Covariance between user rating data can be used to identify users with similar preferences.

Example:
In a movie recommendation system, the covariance between the rating histories of User A and User B is used to compute similarity, enabling personalized content suggestions based on shared preferences.

import numpy as np

# User evaluation data
user_A = [5, 3, 4, 4]
user_B = [3, 1, 2, 3]

cov_matrix = np.cov(user_A, user_B, bias=True)
print(cov_matrix)

5. Detection of Quantum Entanglement (Quantum Information Theory)

Background:
Covariance matrices are used to evaluate entanglement between qubits, serving as one method to determine whether a quantum state exhibits entanglement.

Example:
By computing the covariance matrix of two qubits and checking the positivity of the partial transpose, one can determine whether the state is entangled.

6. Image Recognition (Computer Vision)

Background:
Covariance is used in feature extraction and pattern recognition tasks in image processing.

Example:
In face recognition algorithms such as Eigenfaces, covariance matrices are used to extract principal facial features from image datasets.

7. Time Series Analysis

Background:
Covariance is employed to analyze the relationship between observations at different time points.

Example:
Applications include analyzing the correlation between temperature and humidity, stock price movements, and foreign exchange rate fluctuations.

Summary

Covariance is not just a tool to measure the relationship between two variables—it plays a key role across a wide range of applications, including data compression, anomaly detection, signal processing, and even analyzing entangled quantum states in quantum mechanics.

Recommended Books and Papers on Quantum Entanglement and Covariance

1. Foundational Theories

Quantum Mechanics Fundamentals

• Title: Quantum Mechanics and Path Integrals

• Authors: Richard P. Feynman, Albert R. Hibbs

• Publisher: Dover Publications

• Overview:
  A foundational book on quantum mechanics.
  It provides a detailed explanation of probability amplitudes and wave functions, which is essential for understanding quantum entanglement.

Introduction to Quantum Information Theory

• Title: Quantum Computation and Quantum Information

• Authors: Michael A. Nielsen, Isaac L. Chuang

• Publisher: Cambridge University Press

• Overview:
  Covers the basic concepts of quantum computing and information.
  Topics include quantum entanglement, quantum teleportation, and Bell’s inequalities.

2. Covariance and Quantum Entanglement

Covariance Matrices in Quantum Information Science

• Title:Quantum Information With Continuous Variables of Atoms and Light

Gaussian Quantum Information and Covariance

• Title: Quantum Information with Continuous Variable systems: Gaussian and non-Gaussian resources for quantum communications

3. Practical Applications and Computational Techniques

Quantum Algorithms and Covariance Matrices

• Title: Quantum Algorithms via Linear Algebra

• Authors: Richard J. Lipton, Kenneth W. Regan

• Publisher: MIT Press

• Overview:
  Provides numerical methods for quantum algorithms.
  Useful for analyzing quantum states using covariance matrices and verifying Bell inequalities.

Experimental Verification of Quantum Entanglement

• Title: Experimental Quantum Computation and Information

• Authors: Serge Haroche, Jean-Michel Raimond

• Publisher: Oxford University Press

• Overview:
  Focuses on experimental proof of quantum entanglement.
  Includes practical methods for calculating covariance matrices and error analysis in experiments.

4. Key Research Papers

Bell’s Inequality and Covariance

• Title: Bell’s Theorem, Quantum Theory and Conceptions of the Universe

• Author: M. Kafatos

• Publisher: Springer

• Overview:
  Interprets Bell’s inequality and quantum entanglement from the perspective of covariance matrices.
  Theoretical exploration of non-locality versus classical probability theory.

Continuous-Variable Entanglement Criteria

• Title: Entanglement Criteria for Continuous Variable Systems

• Author: R. Simon

• Journal: Physical Review Letters

• Overview:
  Proposes entanglement criteria using covariance matrices.
  Includes mathematical proof of the PPT (Positive Partial Transpose) criterion.

5. Numerical Simulation

Quantum Computing with Python

• Title: Learn Quantum Computing with Python and IBM Quantum Experience: A hands-on introduction to quantum computing and writing your own quantum programs with Python

• Authors: Sarah C. Kaiser, Christopher Granade

• Publisher: Packt Publishing

• Overview:
  Introduces quantum computing and simulation using Python.
  Includes examples of covariance matrix computation using Qiskit and NumPy.