Finite-Dimensional Quantum Skeletons

Abstract

Finite-dimensional Pauli, no-character, and trace-probability skeletons.

D5/L/Quantum/busch2003effectprobabilities

For the effect form of the finite-dimensional representation theorem, assume a single probability function on every positive contraction, normalized at the identity and additive whenever the sum is again a positive contraction. A single function on effects expresses the required independence from measurement context. These are additional probability assumptions; neither a fixed event structure nor a positive history length supplies them.

Binary additivity first gives the zero value at zero and monotonicity. Dividing an effect into finitely many equal summands gives rational homogeneity. Squeezing a real scalar between rational scalars then gives real homogeneity on the unit interval. For a positive matrix, divide by a positive scalar larger than its norm, evaluate the resulting effect, and multiply by the scalar. Homogeneity proves independence of the chosen scale; a common scale proves additivity on the positive cone.

A Hermitian matrix is a difference of positive matrices. Equality of two such differences gives equality of the cross sums, so the positive-cone functional extends unambiguously to a real-linear functional on Hermitian matrices and then to a complex-linear functional on all matrices. Finite-dimensional trace duality gives a unique representing matrix. Testing positive rank-one matrices proves that this matrix is positive, and evaluation at the identity proves trace one. This argument uses only finite additivity and includes dimension two. Busch gives the effect-measure argument; countable additivity is needed for normality in his infinite-dimensional formulation. The rank-one trace pairing specializes to the squared modulus of the inner product of unit vectors.

Theorem 1.1 (The standard qubit Weyl pair has the Pauli star skeleton).

$ZX=-(XZ)\wedge X^{\ast }=X\wedge Z^{\ast }=Z\wedge X^2=I\wedge Z^2=I$

Proof. Machine-checked in Lean as D5/S3/Quantum/FiniteDimensional.qubit_weyl_star (✓ std3). ∎

Citation. Julian Schwinger (1960). Unitary Operator Bases. DOI: 10.1073/pnas.46.4.570.

Commentary.

The standard two-dimensional Pauli X and Z matrices anticommute, are self-adjoint, and square to the identity. This is only the d = 2 Weyl specialization: it does not identify an arbitrary observer window with a full matrix algebra, prove prime-power tensor factorization or a general qudit relation, or derive the structure from a classical ontology. Original numerical-certificate claim not formalized: the source atom’s matrix-unit relations with exact zero certificate error.

Theorem 1.2 (The qubit matrix algebra has no complex-algebra character).

$\mathrm{IsEmpty}(\mathrm{QubitMatrix}{\to }_{\mathbb{C}\text{-alg}}\mathbb{C})$

Proof. Machine-checked in Lean as D5/S3/Quantum/FiniteDimensional.qubit_matrix_algebra_has_no_character (✓ std3). ∎

Citation. Gerard J. Murphy (1990). C-Algebras and Operator Theory*. DOI: 10.1016/C2009-0-22289-6.

Commentary.

No unital complex-algebra homomorphism from the two-by-two full matrix algebra to the complex numbers exists. The proof uses the stronger global additive and multiplicative laws of an algebra character. D5/L/kochen1968problem is contextual background only: this declaration is not the Kochen-Specker valuation theorem, does not exclude qubit noncontextual projection valuations, and proves neither the arbitrary M_n result for n greater than one nor any CHSH, hidden-address, or probability-is-not-ignorance claim.

Theorem 1.3 (Positive trace-one matrices give nonnegative projection weights).

$$\forall n[\mathrm{Fintype}(n)][\mathrm{DecidableEq}(n)],\forall \rho \in M_n(\mathbb{C}),\mathrm{PosSemidef}(\rho )\wedge \mathrm{tr}(\rho )=1\Rightarrow \mathrm{bornProbability}(\rho ,I)=1\wedge (\forall P,Q,\mathrm{bornProbability}(\rho ,P+Q)=\mathrm{bornProbability}(\rho ,P)+\mathrm{bornProbability}(\rho ,Q))\wedge (\forall P,P^{\ast }=P\Rightarrow P^2=P\Rightarrow 0\le \mathrm{bornProbability}(\rho ,P))$$

Proof. Machine-checked in Lean as D5/S3/Quantum/FiniteDimensional.born_probability_skeleton (✓ std3). ∎

Citation. Andrew M. Gleason (1957). Measures on the Closed Subspaces of a Hilbert Space. DOI: 10.1512/iumj.1957.6.56050.

Commentary.

For a positive semidefinite finite complex matrix rho with trace one, P maps to trace(rho P), is normalized at the identity, is additive, and is nonnegative for every self-adjoint idempotent P. Positivity follows from the compression P rho P* and does not assume that rho commutes with P. D5/L/born1926zur records the historical Born context only. The declaration proves no Gleason representation or uniqueness theorem, no rank-one pure-state modulus-square reduction, no ledger-derived noncontextuality, no harmonic or quartic numerical certificate, and no forced classical-to-quantum origin. Original numerical-certificate claim not formalized: the source atom’s separate Born control group balance to 10^-16.

References

• Truth anchor: D5/S3/Quantum/FiniteDimensional.born_probability_skeleton
• Truth anchor: D5/S3/Quantum/FiniteDimensional.qubit_matrix_algebra_has_no_character
• Truth anchor: D5/S3/Quantum/FiniteDimensional.qubit_weyl_star