From Entangled Clocks to Local Realism and Hidden Variables
Why perfect entanglement correlations naturally suggest predetermined answers, what locality and realism mean, and why Einstein was led to the idea of hidden variables.
Read note →Journal
Tag: quantum-computing
Long-form notes, Qiskit experiments, and applied optimization thinking.
Using the CNOT Gate to Entangle Qubits
A step-by-step study of Bernhardt's example showing how an initially unentangled pair of qubits becomes the Bell state through a CNOT gate, beginning with the tensor product and following the calculation explicitly.
Read note →Making Nullspace and Linear Independence Intuitive: Redundancy and Invisible Input Directions
A question-driven explanation of nullspace and linear independence, moving beyond Ax = 0 to the mental pictures of redundant vectors and input changes that a matrix cannot detect.
Read note →Vector Spaces and Subspaces: Why Linear Combinations Must Stay Inside
Beginning Chapter 3 with an intuitive understanding of vector spaces and subspaces: real components, R^2, closure under linear combinations, why y=2x forms a subspace but y=2x+1 does not, and how constraints reduce independent freedom.
Read note →Understanding Transposes and Permutations: Why Rows and Columns Change Roles
A practical path through transpose and permutation matrices: why transpose is needed, how it connects rows and columns, why (AB)^T reverses order, why permutation matrices satisfy P^{-1}=P^T, and how left multiplication PA actually rearranges rows.
Read note →Where Did the Qubit Superposition Equation Come From?
The familiar qubit equation |ψ⟩ = α|0⟩ + β|1⟩ was not invented as a standalone formula by one person. This note follows the question of where it came from, connecting classical wave superposition, Schrödinger's linear wave mechanics, Dirac's abstract state notation, and the later language of qubits.
Read note →Measuring Entangled Qubits in Same and Different Bases
Entanglement becomes much clearer when we stop treating it only as a factorization test and start asking what Alice and Bob actually observe when they measure. This note develops the Bell-state measurement behavior step by step, showing why same-basis measurements are perfectly correlated while different-basis measurements become 50/50.
Read note →Tensor Products and Entanglement: Why Two Qubits Need a Joint State
Tensor products answer a basic question in quantum mechanics: how do we describe two quantum systems together? This note develops tensor products from scratch, shows why two qubits require a four-dimensional joint state space, and explains why a pure two-qubit state is entangled when it cannot be factored into separate pure states for Alice and Bob.
Read note →