Redshift Academy

Wolfram Alpha:         

  Search by keyword:  

Astronomy

-
Astronomical Distance Units .
-
Celestial Coordinates .
-
Celestial Navigation .
-
Location of North and South Celestial Poles .

Chemistry

-
Avogadro's Number .
-
Balancing Chemical Equations
-
Stochiometry
-
The Periodic Table .

Classical Mechanics

-
Blackbody Radiation .

Classical Physics

-
Archimedes Principle
-
Bernoulli Principle
-
Center of Mass Frame
-
Comparison Between Gravitation and Electrostatics
-
Compton Effect .
-
Coriolis Effect
-
Cyclotron Resonance
-
Dispersion
-
Doppler Effect
-
Double Slit Experiment
-
Elastic and Inelastic Collisions .
-
Electric Fields
-
Error Analysis
-
Fick's Law
-
Fluid Pressure
-
Gauss's Law of Universal Gravity .
-
Gravity - Force and Acceleration
-
Hooke's law
-
Ideal and Non-Ideal Gas Laws (van der Waal)
-
Impulse Force
-
Inclined Plane
-
Inertia
-
Kepler's Laws
-
Kinematics
-
Kinetic Theory of Gases .
-
Kirchoff's Laws
-
Maxwell's Equations .
-
Moments and Torque
-
Nuclear Spin
-
One Dimensional Wave Equation .
-
Pascal's Principle
-
Phase and Group Velocity
-
Poiseuille's Law
-
Refractive Index
-
Rotational Dynamics
-
Simple Harmonic Motion
-
Specific Heat, Latent Heat and Calorimetry
-
The Gas Laws
-
The Laws of Thermodynamics
-
The Zeeman Effect .
-
Young's Modulus

Climate Change

-
Keeling Curve .

Cosmology

-
Baryogenesis
-
Cosmic Background Radiation and Decoupling .
-
CPT Symmetries
-
Dark Matter .
-
Friedmann-Robertson-Walker Equations .
-
Hubble's Law
-
Inflation Theory
-
Introduction to Black Holes .
-
Olbers' Paradox .
-
Planck Units .
-
Stephen Hawking's Last Paper .
-
Stephen Hawking's PhD Thesis .
-
The Big Bang Model
-
Vacuum Energy .

Finance and Accounting

-
Amortization
-
Annuities
-
Brownian Model of Financial Markets .
-
Capital Structure
-
Dividend Discount Formula
-
Lecture Notes on International Financial Management
-
NPV and IRR
-
Periodically and Continuously Compounded Interest
-
Repurchase versus Dividend Analysis

Game Theory

-
The Truel .

General Relativity

-
Basis One-forms .
-
Catalog of Spacetimes .
-
Curvature and Parallel Transport
-
Einstein's Field Equations
-
Geodesics
-
Gravitational Waves
-
Hyperbolic Motion and Rindler Coordinates .
-
Quantum Gravity
-
Ricci Decomposition .
-
Ricci Flow .
-
Stress-Energy-Momentum Tensor .
-
Tensors
-
The Area Metric
-
The Dirac Equation in Curved Spacetime .
-
The Equivalence Principal
-
The Essential Mathematics of General Relativity
-
The Induced Metric
-
The Light Cone .
-
The Metric Tensor .
-
The Principle of Least Action in Relativity .
-
Vierbein (Frame) Fields

Group Theory

-
Basic Group Theory .
-
Basic Representation Theory .
-
Building Groups From Other Groups .
-
Sets, Groups, Modules, Rings and Vector Spaces
-
Symmetric Groups .
-
The Integers Modulo n Under + and x .

Lagrangian and Hamiltonian Mechanics

-
Classical Field Theory .
-
Euler-Lagrange Equation .
-
Ex: Newtonian, Lagrangian and Hamiltonian Mechanics .
-
Hamiltonian Formulation .
-
Liouville's Theorem
-
Symmetry and Conservation Laws - Noether's Theorem .

Macroeconomics

-
Lecture Notes on International Economics
-
Lecture Notes on Macroeconomics
-
Macroeconomic Policy

Mathematics

-
Amplitude, Period and Phase
-
Arithmetic and Geometric Sequences and Series .
-
Asymptotes
-
Augmented Matrices and Cramer's Rule
-
Binomial Theorem (Pascal's Triangle)
-
Completing the Square
-
Complex Numbers
-
Composite Functions
-
Conformal Transformations .
-
Conjugate Pair Theorem
-
Contravariant and Covariant Components of a Vector
-
Derivatives of Inverse Functions
-
Double Angle Formulas
-
Eigenvectors and Eigenvalues
-
Euler Formula for Polyhedrons
-
Factoring of a3 +/- b3
-
Fourier Series and Transforms .
-
Fractals
-
Gauss's Divergence Theorem
-
Grassmann and Clifford Algebras .
-
Heron's Formula
-
Index Notation (Tensors and Matrices) .
-
Inequalities
-
Integration By Parts
-
Introduction to Conformal Field Theory .
-
Inverse of a Function
-
Law of Sines and Cosines
-
Line Integrals, ∮
-
Logarithms and Logarithmic Equations
-
Matrices and Determinants
-
Matrix Exponential
-
Mean Value and Rolle's Theorem
-
Modulus Equations
-
Orthogonal Curvilinear Coordinates .
-
Parabolas, Ellipses and Hyperbolas
-
Piecewise Functions
-
Polar Coordinates
-
Polynomial Division
-
Quaternions 1 .
-
Quaternions 2 .
-
Regular Polygons
-
Related Rates
-
Similar Matrices and Diagonalization .
-
Spherical Trigonometry
-
Stirling's Approximation
-
Sum and Differences of Squares and Cubes
-
Symbolic Logic
-
Tangent and Normal Line
-
Taylor and Maclaurin Series .
-
The Essential Mathematics of Lie Groups
-
The Limit Definition of the Exponential Function
-
Tic-Tac-Toe Factoring
-
Trapezoidal Rule
-
Unit Vectors
-
Volume Integrals

Microeconomics

-
Marginal Revenue and Cost

Nuclear Physics

-
-
Radioactive Decay

Particle Physics

-
Feynman Diagrams and Loops
-
Field Dimensions
-
Helicity, Chirality and Weyl Spinors .
-
Klein-Gordon and Dirac Equations .
-
Regularization and Renormalization
-
Scattering - Mandelstam Variables
-
Spin 1 Eigenvectors .

Probability and Statistics

-
Box and Whisker Plots
-
Buffon's Needle .
-
Categorical Data - Crosstabs
-
Chebyshev's Theorem
-
Chi Squared Goodness of Fit
-
Conditional Probability
-
Confidence Intervals
-
Data Types
-
Expected Value
-
Factor Analysis
-
Hypothesis Testing
-
Linear Regression
-
Monte Carlo Methods
-
Non Parametric Tests
-
One-Way ANOVA
-
Pearson Correlation
-
Permutations and Combinations
-
Pooled Variance and Standard Error
-
Probability Distributions
-
Probability Rules
-
Sample Size Determination
-
Sampling Distributions
-
Set Theory - Venn Diagrams
-
Stacked and Unstacked Data
-
Stem Plots, Histograms and Ogives
-
Survey Data - Likert Item and Scale
-
Tukey's Test
-
Two-Way ANOVA

Programming and Computer Science

-
Hashing
-
How this site works ...
-
More Programming Topics
-
MVC Architecture
-
Open Systems Interconnection (OSI) Standard - TCP/IP Protocol
-
Public Key Encryption

Quantum Computing

-
Density Operators and Mixed States .
-
Entangled States .
-
The Qubit .

Quantum Field Theory

-
Creation and Annihilation Operators
-
Field Operators for Bosons and Fermions
-
Lagrangians in Quantum Field Theory
-
Path Integral Formulation
-
Relativistic Quantum Field Theory

Quantum Mechanics

-
Bohr Atom
-
Clebsch-Gordan Coefficients .
-
Commutators
-
Dyson Series
-
Electron Orbital Angular Momentum and Spin
-
Heisenberg Uncertainty Principle
-
Ladder Operators .
-
Multi Electron Wavefunctions .
-
Pauli Spin Matrices
-
Photoelectric Effect .
-
Position and Momentum States .
-
Probability Current
-
Schrodinger Equation for Hydrogen Atom .
-
Schrodinger Wave Equation
-
Spin 1/2 Eigenvectors
-
The Differential Operator
-
The Essential Mathematics of Quantum Mechanics
-
The Quantum Harmonic Oscillator .
-
The Schrodinger, Heisenberg and Dirac Pictures
-
The WKB Approximation
-
Time Dependent Perturbation Theory
-
Time Evolution and Symmetry Operations
-
Time Independent Perturbation Theory
-
Wavepackets

Semiconductor Reliability

-
The Weibull Distribution

Solid State Electronics

-
Band Theory of Solids .
-
Fermi-Dirac Statistics .
-
Intrinsic and Extrinsic Semiconductors .
-
The MOSFET
-
The P-N Junction

Special Relativity

-
4-vectors .
-
Electromagnetic (Faraday) Tensor .
-
Energy and Momentum in Special Relativity, E = mc2 .
-
Invariance of the Velocity of Light .
-
Lorentz Invariance .
-
Lorentz Transform .
-
Lorentz Transformation of the EM Field .
-
Newton versus Einstein
-
Spinors - Part 1 .
-
Spinors - Part 2 .
-
The Continuity Equation .
-
The Lorentz Group .

Statistical Mechanics

-
Entropy and the Partition Function
-
The Harmonic Oscillator
-
The Ideal Gas

String Theory

-
Bosonic Strings
-
Extra Dimensions
-
Introduction to String Theory
-
Kaluza-Klein Compactification of Closed Strings
-
Strings in Curved Spacetime
-
Toroidal Compactification

Superconductivity

-
Bardeen–Cooper–Schrieffer Theory
-
BCS Theory
-
Cooper Pairs
-
Introduction to Superconductivity .
-
Superconductivity (Lectures 1 - 10)
-
Superconductivity (Lectures 11 - 20)

Supersymmetry (SUSY) and Grand Unified Theory (GUT)

-
Chiral Superfields
-
Generators of a Supergroup
-
Grassmann Numbers
-
Introduction to Supersymmetry
-
The Gauge Hierarchy Problem

The Standard Model

-
Electroweak Unification (Glashow-Weinberg-Salam)
-
Gauge Theories (Yang-Mills)
-
Gravitational Force and the Planck Scale
-
Introduction to the Standard Model
-
Isospin, Hypercharge, Weak Isospin and Weak Hypercharge
-
Quantum Flavordynamics and Quantum Chromodynamics
-
Special Unitary Groups and the Standard Model - Part 1 .
-
Special Unitary Groups and the Standard Model - Part 2
-
Special Unitary Groups and the Standard Model - Part 3 .
-
Standard Model Lagrangian
-
The Higgs Mechanism
-
The Nature of the Weak Interaction

Topology

-

Units, Constants and Useful Formulas

-
Constants
Last modified: January 19, 2022 ✓

Projection Operator Representation ---------------------------------- A projection operator projects a vector onto a sub space. Note: The projection operator produces another vector and is not the same as the dot product, u.v = |u||v|cosθ, which produces a scalar. Example: Consider the following: The vector, u, is defined in 3D space. We want to find the linear operator that returns the projection of this vector onto the subspace with the orthonormal basis of {x,y}. - - - - | 1 | | 0 | x = | 0 | y = | 1 | | 0 | | 0 | - - - - The projection operator that projects u onto the x-y plane is: ℙxy := Σ|n><n| where n are the subspace orthogonal n basis vectors. So, ℙ = |x><x| + |y><y| - - - - - - - - | 1 || 1 0 0 | + | 0 || 0 1 0 | = | 0 | - - | 1 | - - | 0 | | 0 | - - - - - - - - | 1 0 0 | + | 0 0 0 | = | 0 0 0 | | 0 1 0 | | 0 0 0 | | 0 0 0 | - - - - - - | 1 0 0 | = | 0 1 0 | | 0 0 0 | - - - - So clearly, if have a vector | x | then the | y | | z | - - projection onto the x-y plane is: - - - - - - | 1 0 0 | | x | | x | | 0 1 0 | | y | = | y | as expected | 0 0 0 | | z | | 0 | - - - - - - If the space and subspace are the same, the projection operator is the unitary matrix, I, and the vector returned is just the original vector. Properties of ℙ --------------- Idempotent: The projection operator satisfies the operator equation ℙℙ = ℙ. This means that acting twice with a projection operator on a vector gives the same result as acting once. ℙ2 = (|ψ><ψ|)(|ψ><ψ|) = |ψ><ψ|ψ><ψ| = |ψ><ψ| if <ψ|ψ> = 1 (normalized) ℙ|ψ> = λ|ψ> ℙ2|ψ> = λ2|ψ> This result implies that the eigenvalues of ℙ can only be 0 or 1. Hermitian: ℙ = (|ψ><ψ|) = (|ei><ei|)(|ei><ei|) = |ei> (<ei|ei>) <ei| = |ei> 1 <ei| = |ei><ei| = ℙ Density Matrix Representation ----------------------------- A density matrix is an alternative way to describe the quantum state of system. Instead of using state vectors or wavefunctions, the states are represented by matrices. Wavefunctions can only represent pure states while density matrices can be used to represent both pure and mixed states. Pure States ----------- The density matrix, ρ, for a pure state is simply the projection operator, i.e. ρ = |ψ><ψ| = Σαiαj*|ei><ej| ijij = <ei|ρ|ei> = <ei|ψ><ψ|ei> = aiaj*] Which is just the projection operator. A density operator represents a pure state if and only if: 1. ρ2 = ρ 2. Tr(ρ) = 1 3. Tr(ρ2) = 1 (Purity) Consider the pure state: |ψ> = 1/√2(|u> + |d>) ρ = |ψ><ψ| = (1/2)(|u><u| + |u><d| + |d><u| + |d><d|) - - = | 1/2 1/2 | | 1/2 1/2 | - - Note: The cross terms remain because >< is the tensor product, not the dot product, i.e. - - - - <u|d> = | 1 0 || 0 | = 0 - - | 1 | - - - - - - - - |u><d| = | 1 || 0 1 | = | 0 1 | | 0 | - - | 0 0 | - - - - The non-zero off-diagonal elements represen the interference (coherence) terms associated with quantum superposition. Density Matrix for the Singlet ------------------------------ Matrix Vectorization: Vectorization of a matrix is a linear transformation which converts the matrix into a column vector. Specifically, the vectorization of a m × n matrix A, denoted vec(A), is the mn × 1 column vector obtained by stacking the columns of the matrix A on top of one another. Example: - - - - u = | 1 | d = | 0 | | 0 | | 1 | - - - - - - - - | a | vec | a b | = | c | | c d | | b | - - | d | - - - - - - - - | 1 | |uu> ≡ |u ⊗ u> = | 1 0 | vec | 1 0 | => | 0 | | 0 0 | | 0 0 | | 0 | - - - - | 0 | - - We can apply vectorization of a matrix to the Singlet state. Therefore, - - - - - - | 0 | (1/√2)| 1 | ⊗ | 0 | = | 1/√2 | ... (1) | 0 | | 1 | | 0 | - - - - | 0 | - - - - - - - - | 0 | (1/√2)| 0 | ⊗ | 1 | = | 0 | ... (2) | 1 | | 0 | | 1/√2 | - - - - | 0 | - - - - | 0 | (1) - (2) = | 1/√2 | = ψ | -1/√2 | | 0 | - - - - | 0 | - - ρ = |ψ><ψ| = | 1/√2 || 0 1/√2 -1/√2 0 | | -1/√2 | - - | 0 | - - - - | 0 0 0 0 | = | 0 1/2 -1/2 0 | | 0 -1/2 1/2 0 | | 0 0 0 0 | - - ρ2 = ρ therefore this is a pure entangled state. We know this is an entangled state because it cannot be factored but can we determine this by just looking at ρAB? The answer is yes but we first have to construct the REDUCED DENSITY MATRIX. This is discussed in the next section. Density Operator for Mixed States --------------------------------- A mixed state is a statistical ensemble of normalized pure states. It is not a superposition and because it is statistical in nature, there is limited or no knowledge about the state of the system the state represents. Unlike a pure state, which can be unequivocally defined by the single state vector, |ψ>, which has complete information about the system, a mixed state cannot be described by a wave function. Instead, it requires a density matrix, ρ, for its description. The density operator for a mixed state is: ρ = ΣPkρk| where ΣPk = 1 and ρk is for a pure state. k It satisfies: 1. ρ2 ≠ ρ 2. Tr(ρ) = 1 3. Tr(ρ2) ≠ 1 With a coherent superposition there is a well defined phase relationship between the pieces in the superposition, which means that there can be interference between these pieces in subsequent operations. Consider the state: |ψ> = 1/√2(exp(iθ)|d> + exp(iφ)|u>) If the phase, (θ - φ), is constant, the density matrix for this state looks like: - - ρ = | 1/2 exp(i(θ-φ))/2 | | exp(-i(θ-φ))/2 1/2 | - - Since ρ2 = ρ, this state is pure. Now, suppose that (θ - φ) is random (i.e. the phase difference between |u> and |d> is random). In this case the expectation value of exp(i(θ-φ)) is 0. The density matrix becomes: - - ρ' = | 1/2 0 | | 0 1/2 | - - ρ no longer represents a pure state. This is simply a classical statistical probability law. The off-diagonal elements of ρ have disappeared meaning that all the coherence has been lost resulting in a completely mixed state. It is also possible to have a partially mixed state where the off-diagonal elements of the density matrix are not equal to 0. This indicates that the state is a combination consisting of a statistical ensemble of basis elements and a quantum superposition of the same basis elements. In other words, the off-diagonal entries represent the coherences in the mixed state. Note that the expectation value of an operator corresponding to an observable acting on a state defined by ρ is given by <O> = Tr(ρO). Since the diagonal elements of ρ are the same in each case, there is no way to physically distinguish a coherent superposition from a mixed state. The only way to tell is if an operator exists that satisfies the definition of a coherent superposition as mentioned before. For example, <σx> = Tr(ρσx) gives: - - - - - - Tr| 1/2 1/2 || 0 1 | = Tr| 1/2 1/2 | = 1 | 1/2 1/2 || 1 0 | | 1/2 1/2 | - - - - - - - - - - - - Tr| 1/2 0 || 0 1 | = Tr| 0 1/2 | = 0 | 0 1/2 || 1 0 | | 1/2 0 | - - - - - - Mixed states are important in the fields of quantum computing and communication where there maybe noise in the system that could cause a state to flip with some probability, p. For example, Alice could send Bob the state |u> and Bob could end up with the state |d> with probability, p, or the desired state with the probability, 1 - p. So Bob's state is either |u> or |d> but not a quantum superposition of the two. A more practical example is the qubit. Suppose we want to prepare a system in the following coherent state: |ψ1> = (1/√2)|u> + (1/√2)|d> However, because of noise in the system, there is only an 80% chance of achieving this. 10% of the time we get: |ψ2> = (1/2)|u> + (√3/2)|d> and 10% we get: |ψ3> = (√3/2)|u> + (1/2)|d> Since we do not know the outcome of our qubit every time we prepare it, we can represent it as a mixed state of the form: ρ = (4/5)|ψ1><ψ1| + (1/5)|ψ2><ψ2| + (1/5)|ψ3><ψ3| Where 4/5, 1/5 and 1/5 are the classical probabilities of obtaining the states |ψ1>, |ψ2> and |ψ3>. - - ρ = | 1/2 √3/20 + 2/5 | | √3/20 + 2/5 1/2 | - - The matrix tells us that the state is not only a mixed state (ρ2 ≠ ρ and Tr(ρ2) < 1) but also has some level of coherent superposition. The ratio coherence/mixed is dependent on the amount of coherence 'leaked' off due to interactions with the local environment. Reduced Density Matrix ---------------------- The reduced density operator describes completely all the properties/outcomes of measurements of the subsystem B, given that system A is left unobserved (”tracing out”) subsystem A. ρA = TrB(|ψi><ψj| ⊗ |φi><φj|) = |ψi><ψj|Tr(|φi><φj|) = |ψi><ψj|<φij>) For the singlet: ρAB = |ψ><ψ| = (1/2)(|uAdB> - |dAuB>)(<uAdB| - <dAuB|) = (1/2)(|uAdB><uAdB| - |uAdB><dAuB| - |dAuB><uAdB| + |dAuB><dAuB|) = (1/2)((|uA><uA|)|dB><dB| - (|uA><dA|)|dB><uB| - (|dA><uA|)|uB><dB| + (|dA><dA|)|uB><uB|) ρB = TrAAB) ρB = (1/2)(Tr(|uA><uA|)|dB><dB| - Tr(|uA><dA|)|dB><uB| - Tr(|dA><uA|)|uB><dB| + Tr(|dA><dA|)|uB><uB|) = (1/2)((<uA|uA>)|dB><dB| - (<uA|dA>)|dB><uB| - (<dA|uA>|)|uB><dB| + (<dA|dA>)|uB><uB|) = (1/2)(|dB><dB| + |uB><uB|) - - = | 1/2 0 | | 0 1/2 | - - Likewise, - - ρA = | 1/2 0 | | 0 1/2 | - - In this specific example, ρA and ρB are equal, but this is not always the case. This says that the subsystems A and B are both in mixed states because Tr(ρA) ≠ Tr(ρA2) and Tr(ρB) ≠ Tr(ρB2). This result might seem rather strange since we started out with a pure entangled state. This reflects the fact that an observer who doesn't know about the entanglement (cut off from B) will describe A as a mixed state, while an observer who knows about the entanglement with B will describe them as entangled. Therefore, in ignoring part of the information about the state we are forced to deal with a mixed state. In fact the state is said to be entangled only if the reduced density matrix describes a mixed state. Otherwise, the state is separable. The reduced density matrix is used to quantify the amount of entanglement in a system in terms of its entropy. Summary:     | Density Op | State Vector    ---------------+------------------+--------------------- Description    | ρ | |ψ>    ---------------+------------------+--------------------- Expection(1) | <O> = Tr(ρO) | <O> = <ψ|O|ψ>    ---------------+------------------+--------------------- Probability(2) | P(λ) = Tr(ρℙ) | P(λ) = <ψ|ℙ|ψ>    ---------------+------------------+--------------------- Eigenvalue    | | O|ψ> = λ|ψ>    ---------------+------------------+--------------------- Separability(3)| ρAB = ΣPkρA ⊗ ρB | ψAB = ψA ⊗ ψB     | k |    ---------------+------------------+--------------------- Reduced(5) | ρA = TrBAB )  |    ---------------+------------------+--------------------- TDSE(6) | (1/ih)[H,ρ] | d/dt|ψ> = (1/ih)H|ψ>    ---------------+------------------+--------------------- 1. Tr(ρO) = Σ<ei|ρO|ei> by definition i = Σ<ei|ρ|ei><ei|O|ei> i = ΣρiiOii i = Tr(ρO) 2. Density Operator: - - - - - - Tr(| 1/2 1/2 || 1 0 |) = Tr(| 1/2 0 |) = 1/2 | 1/2 1/2 || 0 0 | | 1/2 0 | - - - - - - State Vector: - - - - - - <u|ℙ|u> = | 1/√2 0 || 1 0 || 1/√2 | - - | 0 0 || 0 | - - - - - - - - = | 1/√2 0 || 1/√2 | = 1/2 - - | 0 | - - In general: - - - - - - | α β || 1 0 || α | = αα* - - | 0 0 || β | - - - - 3. Pure state: For pure state Pk = 1. |ψ> = (1/√2)(|u> + |d>) ⊗ (1/√2)(|u> + |d>): - - | 1/4 1/4 1/4 1/4 | - - | 1/4 1/4 1/4 1/4 | = | 1/2 1/2 | ⊗ | 1/2 1/2 | | 1/4 1/4 1/4 1/4 | | 1/2 1/2 | r | 1/2 1/2 | | 1/4 1/4 1/4 1/4 | - - - - Mixed state: ρ = (1/4)((1/√2)|u> + (√1/2)|d>) ⊗ (3/4)((1/√2)|u> + (1/√2)|d>): - - | 1/4 1/4 1/4 1/4 | - - - - | 1/4 1/4 1/4 1/4 | = (1/4)| 1/2 1/2 | ⊗ | 1/2 1/2 | | 1/4 1/4 1/4 1/4 | | 1/2 1/2 | | 1/2 1/2 | | 1/4 1/4 1/4 1/4 | - - - - - - - - - - + (3/4)| 1/2 1/2 | ⊗ | 1/2 1/2 | | 1/2 1/2 | | 1/2 1/2 | - - - - - - | 1/4 1/4 1/4 1/4 | = | 1/4 1/4 1/4 1/4 | | 1/4 1/4 1/4 1/4 | | 1/4 1/4 1/4 1/4 | - - 5. ρA = TrBAB) = TrBA ⊗ ρB) = ρATrBB) = ρA 6. ihd/dt|ψ> = H|ψ> ihd/dt|ρ> = ihd/dt|ψ><ψ| = (d/dt|ψ>)<ψ| + |ψ>(<d/dt|<ψ|) = (1/ih)H|ψ><ψ| - (1/ih)|ψ><ψ|H = (1/ih)Hρ - (1/ih)ρH = (1/ih)[H,ρ] Examples: | Not entangled | Entangled -----+-----------------------------+------------------------- Pure | |ψ> = (1/√2)(|uu> + |du>) | |ψ> = (1/√2)(|ud> - |du>) -----+-----------------------------+------------------------- Mixed| ρ = (1/4)|uu> + (3/4)|du> | ρ = (1/4)|ud> - (3/4)|du>) -----+-----------------------------+-------------------------- Entanglement Entropy -------------------- How do we quantify the degree of entanglement between 2 subsystems constituting a two-part composite quantum system? One way to do this to compute the QUANTUM ENTROPY (von Neumann) defined as: S = -Tr(ρlogρ) = -Σλilogλi i Where λi are the eigenvalues of ρ In classical mechanics this is equivalent to: S = -ΣiPilogPi The entropy is measure of our degree of knowledge about the state of the system. In the case where there is only one state we know exactly what state the system is in and the entropy is 0. Conversely, if the number of states is large, then we have very little knowledge about the precise state of the system and the entropy is large. To compute the entropy of an entangled state it is only necessary to look at the reduced density matrix. What this means is that if you have a combined system in a pure entangled state and you measure either sub-system (in this case A) alone they will, in general, be described by a mixed state with an associated density matrix. Consider the singlet 1/√2{|ud> - |du>}.   - - ρA = | 1/2 0 | with λ = 1/2 with multiplicity 2   | 0 1/2 |   - - and for the entropy we get: SA = -Σλilogλi i = -{(1/2)log(1/2) + (1/2)log(1/2)} = log(2) It is fairly easy to show that 2 is the maximum entropy ever allowed. Therefore, we have a combined system in a pure entangled state but we have complete ignorance about either of the sub-systems (we have focussed on A here but the same is true for B). Now consider the product state: |ψ> = (1/√2)(|uAuB> + |dAuB>) = (1/√2)(|uA> + |dA> ) ⊗ |uB> ρAB = |ψ><ψ| = (1/2)(|uAuB> + |dAuB>)(<uAuB| + <dAuB|) = (1/2)(|uAuB><uAuB| + |uAuB><dAuB| + |dAuB><uAuB| + |dAuB><dAuB|) = (1/2)(|uA><uA|(|uB><uB|) + |uA><dA|(|uB><uB|) + |dA><uA|(|uB><uB|) + |dA><dA|(|uB><uB|)) ρA = TrBAB) - - = | 1/2 1/2 | | 1/2 1/2 | - - This has eigenvalues, λ = 0 and 1. For the entropy we get: SA = -Σλilogλi i = -{(1)log(1) + (0)log(0)} = 0 Monogamy of Entanglement ------------------------ If 2 electrons, are maximally entangled with each other they cannot be entangled at all with a third electron. In general, there will be a trade off between the amount of entanglement between 1 and 2 and 1 and 3 or 2 and 3. This is referred to as the MONOGAMY OF ENTANGLEMENT. In general, the more parties, the less entanglement between them.