closed-loop-quantum-probability

name

closed-loop-quantum-probability

description

Closed-loop decomposition of quantum probabilities from unitarity — Bargmann invariants as phase-invariant loop quantities, Born rule as quadratic structure from forward/reverse amplitude products. Connects quantum probability to number theory (loop invariants, cyclic groups) and statistics (phase-invariant estimation). Trigger words: closed-loop quantum probability, Bargmann invariant, unitarity, Born rule derivation, quantum interference, phase-invariant, cyclic loop.

Closed-Loop Quantum Probability Decomposition

Overview

Reformulates quantum probability decomposition as a direct consequence of unitarity, where closed loops are fundamental quantum entities and interference arises from distinct loop classes weighted by Bargmann phases.

Core Methodology

Closed-Loop Framework

• Quantum probabilities decompose into sums over closed loops in state space
• Each loop class contributes weighted by its Bargmann phase
• Bargmann invariants emerge naturally as phase-invariant quantities (not independently postulated)

Born Rule from Unitarity

• Born rule reflects quadratic structure from forward × reverse amplitude product
• This product defines the fundamental closed loop
• Cross-terms in interference reinterpreted as contributions from distinct loop classes

Mathematical Structure

• Loop decomposition: P = Σ_L c_L · B_L where B_L = Tr(ρ₁ρ₂...ρₙ) is the Bargmann invariant
• Phase invariance: B_L is invariant under global phase transformations
• Connection to cyclic group structure: n-loop invariants form representation of Z_n

Applications

1. Statistical estimation: Phase-invariant quantities enable robust estimation from noisy measurements
2. Number theory connection: Loop invariants map to characters of finite cyclic groups
3. Quantum algorithms: Loop-based decomposition provides alternative framework for quantum probability computation
4. Interpretability: Demystifies interference as loop-class contributions rather than mysterious cross-terms

Implementation

1. Identify loop structure: Decompose probability amplitude into forward/reverse path pairs
2. Compute Bargmann invariants: B_L = ⟨ψ₁|ψ₂⟩⟨ψ₂|ψ₃⟩...⟨ψₙ|ψ₁⟩
3. Weight by phase: Each loop class weighted by exp(i·Arg(B_L))
4. Sum contributions: P = Σ_L |B_L| · exp(i·Arg(B_L))

Key Insight

Unitarity ⟹ closed-loop decomposition ⟹ Bargmann invariants ⟹ Born rule. The chain is deductive, not axiomatic.

Activation

closed-loop quantum probability, Bargmann invariant, unitarity, Born rule derivation, quantum interference, phase-invariant, cyclic loop, quantum foundations