Care for another ladle of quantum physics? (ladle #1)

First installment of an exploration series on quantum physics and quantum computing fundamentals, designed to make these concepts accessible without sacrificing mathematical rigor. Originally published in French.

From bit to qubit

The article lays the groundwork by distinguishing the classical bit (two exclusive states 0 or 1) from the quantum qubit, described by a state vector |ψ⟩ = α|0⟩ + β|1⟩ in a two-dimensional Hilbert space. The probability amplitudes α and β are complex numbers satisfying the normalization condition |α|² + |β|² = 1. The Bloch sphere provides a geometric representation of any qubit state, with the subtlety that quantum orthogonality translates to a 180° angular separation (not 90°) on this sphere.

Superposition and entanglement

Superposition is the fundamental property distinguishing quantum computing: n qubits simultaneously represent 2ⁿ states, enabling massively parallel processing. Entanglement allows qubits to share correlated states regardless of distance, a phenomenon experimentally confirmed by Alain Aspect’s work and in direct contradiction with Einstein’s intuition.

Key quantum algorithms

Shor’s algorithm (large number factorization in polynomial time via quantum Fourier transform) and Grover’s algorithm (search in O(√N) in unstructured databases, proven optimal) illustrate the potential power of quantum computing. Grover’s amplitude amplification works through iterative application of an oracle and a diffusion operator.

Observables and quantum measurement

An observable is represented by a Hermitian operator whose real eigenvalues correspond to possible measurement outcomes. Measurement causes irreversible collapse of the superposition to an eigenstate. Heisenberg’s uncertainty principle (Δx·Δp ≥ ℏ/2) follows directly from the non-commutativity of certain observable pairs.

Schrödinger’s cat demystified

The article concludes with a reinterpretation of the Schrödinger’s cat paradox through a vector analogy: quantum superposition is mathematically a linear combination, like a vector V = α·eₓ + β·eᵧ in R². The fundamental difference lies in the irreversible collapse of quantum measurement, with no analogue in classical geometry.

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Key takeaways

From bit to qubit: the qubit is a state vector |ψ⟩ = α|0⟩ + β|1⟩ in a 2-dimensional Hilbert space, where α and β are complex probability amplitudes satisfying |α|² + |β|² = 1.
Superposition and exponential power: n qubits simultaneously represent 2ⁿ states (vs. only 1 state among 2ⁿ for n classical bits) — it is the parallel processing, not the number of encodable states, that confers the power.
Quantum entanglement: measuring one qubit instantaneously affects the other regardless of distance — experimentally confirmed (Alain Aspect), in direct contradiction with Einstein's intuition.
Key algorithms: Shor (polynomial-time factorization, RSA threat) and Grover (search in O(√N) vs. O(N), proven optimal) — amplitude amplification via oracle + diffusion operator.
Observables and measurement: an observable is a Hermitian operator (real eigenvalues) acting on the state vector; measurement causes irreversible collapse of the superposition to an eigenstate — Heisenberg's uncertainty principle stems from observable non-commutativity.
Schrödinger's cat demystified: quantum superposition is mathematically a linear combination of basis vectors — analogy with a vector V = α·eₓ + β·eᵧ in R², except that quantum measurement is irreversible (collapse) unlike classical geometric projection.