Quantum Measurement as a Computable Coordinate Transformation: A Topological‐Geometric Perspective

Abstract

We propose a unified framework for quantum measurement by interpreting it as a computable coordinate transformation in a topologically non-trivial state space. Building on the path integral formulation's capacity to encode global topological information and the relational quantum mechanics perspective, we demonstrate how measurement outcomes arise from local coordinate choices in a fiber bundle structure. This approach naturally resolves the tension between geometric locality and topological invariance in quantum systems, while offering concrete computational tools via discrete differential geometry and tensor network representations.

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1. Introduction: Measurement as Coordinate Relativity

The Copenhagen interpretation’s "wavefunction collapse" has long been criticized for its ad hoc separation of quantum system and classical observer. Here, we reframe measurement as:

A dynamical selection of local computational bases (coordinates) in a globally entangled quantum state space,
where:

• Observables correspond to choices of coordinate systems
• Projective measurements map to orthogonal projections in local charts
• Measurement backaction reflects holonomy in the connection

This aligns with the computable coordinate systems paradigm (GitHub/panguojun), where physical laws must be expressible in finitely computable reference frames.

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2. Mathematical Framework

2.1 Quantum State Space as a Fiber Bundle

Let the quantum system be described by a principal bundle ( \mathcal{P}(\mathcal{M}, G, \pi) ):

• Base manifold ( \mathcal{M} ) = Space of measurable observables (e.g., position/momentum spectra)
• Fiber ( G ) = Gauge group encoding degeneracies (e.g., ( U(1) ) for phase freedom)
• Section ( \psi(x) ) = Quantum state in a local coordinate chart

A measurement device selects a local trivialization ( {|e_i(x)\rangle} ), inducing a projection:
[ |\psi\rangle \rightarrow |e_i(x)\rangle \langle e_i(x)|\psi\rangle \quad \text{(Coordinate-dependent collapse)} ]

2.2 Path Integral as a Topological Atlas

The Feynman path integral
[ K(q_f,t_f; q_i,t_i) = \int \mathcal{D}[q(t)] e^{iS[q(t)]/\hbar} ]
becomes a sum over all possible coordinate transitions between initial and final states. Crucially:

• Classical paths = Geodesics in the coordinate chart
• Tunneling paths = Non-trivial transitions between charts
• Topological terms (e.g., ( \theta )-vacua) = Obstructions to global coordinate fixing

2.3 Computable Coordinate Systems

Following panguojun/Coordinate-System, we impose:

1. Finiteness: Measurements correspond to finite-dimensional subalgebras of observables
2. Algorithmicity: Coordinate transformations are Turing-computable functions
3. Locality: Each observer accesses only a causal diamond of the bundle

This constrains measurement outcomes to physically realizable computations.

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3. Key Implications

3.1 Resolution of Measurement Paradoxes

• Wigner’s Friend: Different observers’ coordinate charts overlap but need not agree—consistent with sheaf theory
• Delayed Choice: Retrospective coordinate adjustment (like gauge fixing in Yang-Mills theory)

3.2 Topological Protection of Quantum Information

Systems with topological order (e.g., fractional quantum Hall states) exhibit:

• Measurement-robust degeneracies = Coordinate-independent invariants (Chern numbers)
• Anyonic statistics = Monodromy in coordinate transformations

3.3 Quantum Gravity Interface

In holographic duality (AdS/CFT):

• Boundary measurements = Radial coordinate cutoff in bulk
• Bulk reconstruction = Solving the quantum coordinate problem

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4. Computational Implementation

We outline a lattice-compatible scheme:

Step	Operation	Mathematical Implementation
1	Discretize state space	Cellular decomposition of ( \mathcal{M} )
2	Define local measurement frames	MPS/Tensor network embeddings
3	Simulate path integral	Markov chain Monte Carlo on groupoid
4	Extract observables	Sheaf cohomology computation

(Code examples available at GitHub/panguojun/Coordinate-System adapt these steps for qubit systems.)

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5. Conclusions

By treating quantum measurement as a computable coordinate transformation, we:

1. Unify geometric locality and topological invariance
2. Demystify collapse as an emergent coordinate effect
3. Enable new quantum algorithms via topological coordinate optimization

Future work should formalize the computational complexity classes of quantum coordinate choices—a step toward quantum-aware foundations of computation.