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Gauge Freedom and the Born Rule Extending the Unitary Refinement Program to Generalized Quantum Measurements Michael Tolbert September 2026 Abstract Recent work established a representation theorem for finite projective events, showing that equal-norm refinement, primitive equivalence, cardinal composition, and refinement invariance uniquely induce the Born measure on homogeneous branch spaces. That construction, however, applies only to pure states and idealized projection-valued measurements. This paper devel- ops a generalized refinement framework for mixed states and positive operator-valued measures (POVMs). By combining purification and Naimark dilation, arbitrary finite-dimensional open- system descriptions are embedded into enlarged Hilbert spaces in which the original cardinality representation theorem applies. The central result is a Representation Invariance Theorem: physically equivalent purification-dilation realizations induce identical abundance measures de- spite differing ancillary constructions. The result resolves the representation problem rather than merely importing an already-assumed probability rule into the enlarged space. This invariance reveals a representational structure analogous to gauge freedom. Ancillary refinements, purifications, and measurement dilations generate equivalence classes of mathemat- ical realizations whose variation leaves operational typicality unchanged. The generalized Born measure T (E | ρ) = Tr(ρE) is obtained by first applying the projective refinement theorem as a pre-probabilistic counting construction in an enlarged space and only then proving that the trace rule is its compressed, representation-independent form on the operational pair (ρ,E). A Lipschitz bound expressed through trace distance and operator norm shows that small perturbations of states and effects produce controlled changes in typicality. For a finite family of normalized records with pairwise overlaps bounded by ε, Löwdin symmetric orthogonalization, rank-one subspace perturbation geometry, and the operational Lipschitz estimate yield an explicit dimension-dependent error bound that vanishes with ε. Keywords: Born rule; Everett interpretation; POVM; Naimark dilation; purification; quantum information; typicality; gauge freedom; representation invariance; decoherence. 1 Introduction Closed quantum systems evolve unitarily, while operational quantum theory assigns outcome prob- abilities through the Born rule. For a density operator ρ and a finite POVM{E i } satisfying E i ≥ 0 and P i E i = I, the probability assigned to outcome i is Tr(ρE i ). The formula is indispensable to quantum information theory, but its mathematical success does not by itself explain why this particular measure should govern branch-relative records in an Everettian theory. 1

Gauge Freedom and the Born RuleMichael Tolbert The preceding unitary-refinement program separated two questions that are often conflated. The measure problem asks which quantitative functional represents the relative size or typicality of a branch event. The credence problem asks why that measure should govern rational expectation, self-location, or decision. For finite projective decompositions of pure states, the earlier repre- sentation theorem addressed the first question by constructing equal-norm ancillary refinements. Permutation symmetry rendered the descendants primitive-equivalent; cardinal composition fixed relative abundance on the homogeneous representative space; common-refinement independence removed denominator dependence; and regularity extended the rational construction to arbitrary squared norms [1]. That result was intentionally limited. Open systems are represented by mixed density operators, and general measurements are represented by POVMs whose effects need not be mutually orthog- onal or idempotent. A realistic extension must therefore answer whether the derived measure depends on how a mixed state is purified and whether it depends on how a POVM is realized as a projective measurement in a larger space. The central proposal is that these questions should be answered together. A purification and a Naimark dilation convert an operational pair (ρ,{E i }) into a pure state and a projective decom- position in an enlarged Hilbert space. The earlier finite-abundance theorem can then be applied in that enlarged space. The essential task is to prove that the induced typicality is independent of the auxiliary realization and therefore descends to the operational system. We call this require- ment representation invariance. Ancillary realizations form equivalence classes analogous to gauge orbits: distinct mathematical representatives encode the same operational state-measurement pair and must yield the same coarse typicality. The paper establishes four claims. First, purification and Naimark dilation provide a unified pro- jective realization of every finite-dimensional pair (ρ,{E i }). Second, the cardinality representation theorem induces the generalized Born form in any such realization. Third, the Representation Invariance Theorem proves independence from the chosen minimal realization and extends to non- minimal realizations after restriction to generated subspaces. Fourth, operational Lipschitz bounds and a finite-overlap theorem show that small perturbations of states, effects, and finite coarse record families produce controlled changes in typicality. 2 Operational Data and Realizations 2.1 Mixed states and generalized measurements Let H S be a finite-dimensional complex Hilbert space. A state is a positive operator ρ on H S with Tr(ρ) = 1. A finite generalized measurement is a POVM E ={E i } n i=1 with E i ≥ 0, X i E i = I S .(1) An event associated with A ⊆ {1,...,n} is represented by E A

P i∈A E i . Unlike a projective event, E A need not satisfy E 2 A = E A . The objective is to construct T (A | ρ,E) without assuming at the outset that it equals Tr(ρE A ). 2

Gauge Freedom and the Born RuleMichael Tolbert 2.2 Purification A purification of ρ is a pair (H R ,|φ⟩) with |φ⟩∈H S ⊗H R such that Tr R (|φ⟩⟨φ|) = ρ.(2) Minimal purifications are related by a unitary on the purifying factor; larger purifications are related on the support generated by the state through an isometry or partial isometry [2]. 2.3 Naimark realization A Naimark realization of E is a triple (H K ,W,{P i }) in which W : H S → H K is an isometry and {P i } is a projective decomposition satisfying W † P i W = E i for every i.(3) The projectors obey P i P j = δ ij P i and P i P i = I K . A canonical realization is defined on H S ⊗ C n by W|ψ⟩ = X i p E i |ψ⟩⊗|i⟩, P i = I S ⊗|i⟩⟨i|.(4) The construction does not depend on choosing this canonical model [3, 4]. 2.4 Unified purification-dilation realization Choose a purification and a Naimark isometry. Define |Ψ⟩ = (W ⊗ I R )|φ⟩∈H K ⊗H R , Q i = P i ⊗ I R .(5) Then {Q i } is a PVM and ⟨Ψ|Q i |Ψ⟩ =⟨φ|(E i ⊗ I R )|φ⟩ = Tr(ρE i ).(6) At this stage the trace is an algebraic compression identity, not the definition of typicality. 3 Extended Finite-Abundance Construction To keep the extension self-contained, Table 1 summarizes the structure inherited from Paper 1. The table is not a new axiomatization and does not replace the proofs established there. Table 1: Logical structure inherited from Paper 1. StagePaper 1 premise or resultRole here Pointer eventsA finite coarse algebra of stable record events is idealized by orthogonal projectors. Purification and dilation provide the enlarged projective carrier. 3

Gauge Freedom and the Born RuleMichael Tolbert StagePaper 1 premise or resultRole here Equal-norm refinement Rational squared norms admit equal-norm orthogonal descendants. Enlarged-space outcomes are refined into homogeneous representative sets. Primitive equivalence Permutation-related equal-norm descendants are abundance units. The same premise applies within each realization. Cardinal composition Abundance composes over disjoint collections. This remains the source of finite additivity. Common refinement Equivalent rational refinements give the same parent-event abundance. Paper 2 extends description independence to purifications and dilations. Regular extension Continuity or monotonicity extends rational values to real squared norms. The same extension applies in the projective carrier. Event measure T (P | ψ) =∥Pψ∥ 2 .The induced measure descends to T (E | ρ) = Tr(ρE). Credence boundary Additional epistemic principles are required. The distinction is preserved. 3.1 Rational projective geometry in the enlarged space For |Ψ⟩ and {Q i }, define x i = ∥Q i Ψ∥ 2 . If x i = m i /M is rational, the equal-norm refinement theorem decomposes outcome i into m i mutually orthogonal descendants of squared norm 1/M. The descendants form an M-element homogeneous representative set Ω M . 3.2 Abundance axioms The inherited conditions are normalization, permutation invariance, disjoint cardinal composition, and uniform-refinement invariance. They apply on homogeneous representative spaces, not directly as probability axioms on POVM effects. They uniquely give a M (A) =|A|/M. For an irrational x i =∥Q i Ψ∥ 2 , choose rational sequences q k ↑ x i and s k ↓ x i . Monotonicity yields q k ≤ T (x i )≤ s k .(7) Taking k → ∞ forces T (x i ) = x i . Purification and dilation preserve the squared norm being approximated, so the squeezing argument applies unchanged in the enlarged projective space. Theorem 3.1 (Generalized refinement representation). For every finite-dimensional density oper- ator ρ, finite POVM E = {E i }, and purification-dilation realization (|Ψ⟩,{Q i }), the homogeneous finite-abundance construction assigns T R (i) =∥Q i Ψ∥ 2 = Tr(ρE i ).(8) Proof. The projectors form an exact finite projective decomposition of the normalized pure state. The earlier cardinality theorem applies on rational squared-norm vectors and uniquely yields nor- malized descendant cardinality. Common-refinement invariance makes the rational value indepen- dent of the refinement representation, and regular extension gives T R (i) = ∥Q i Ψ∥ 2 . Compression through W and reduction over H R gives the trace expression. 4

Gauge Freedom and the Born RuleMichael Tolbert 4 Representation Invariance 4.1 Continuity of the Paper 1 framework Paper 1 established a conditional sequence: symmetry identifies primitive-equivalent representa- tives; cardinal composition assigns normalized abundance; common refinement and regular exten- sion induce a measure; and separate epistemic principles may identify that measure with branch- relative credence. Purification and dilation do not replace this construction. They supply a larger pure projective carrier in which it operates. Definition 4.1 (Operational equivalence). A realization R of (ρ,E) consists of a purifying space, a purification |φ R ⟩, a Naimark dilation (H K ,W R ,{P R i }), the unified state |Ψ R ⟩ = (W R ⊗ I R )|φ R ⟩, and Q R i = P R i ⊗ I R . Two realizations are operationally equivalent, written R ∼ R ′ , when both compress to the same ρ and the same outcome-labeled POVM. Definition 4.2 (Generated realization subspace). Define H gen K,R := span{P R i W R |χ⟩ :|χ⟩∈H S , i = 1,...,n},(9) and G R :=H gen K,R ⊗ supp(ρ R ).(10) The generated subspace is a carrier for equal-norm descendants, not itself the object of cardinal evaluation. Principle 4.3 (Operational Representation Principle). An abundance assignment intended to char- acterize (ρ,{E i }) must be constant under equivalences of auxiliary representation that preserve ρ, preserve the labeled effects, and intertwine the generated outcome subspaces. This principle generalizes Common Refinement Independence from alternative homogeneous re- finements to alternative purification-dilation realizations. It is an operational noncontextuality condition, not a consequence of unitary dynamics alone. Lemma 4.4 (Equivalence on generated subspaces). Let R and R ′ realize the same outcome-labeled pair. After restriction to G R and G R ′ , there exists a unitary V :G R →G R ′ such that V|Ψ R ⟩ =|Ψ R ′ ⟩, V Q R i = Q R ′ i V.(11) On unreduced auxiliary spaces, the correspondence is represented by a partial isometry with initial and final spaces G R and G R ′ . Proof. Minimal purifications are related by a unitary on the purifying support. Minimal Naimark dilations of the same labeled POVM are unitarily equivalent on the system-generated subspaces. The tensor product of these unitaries gives V . Nonminimal inaccessible sectors are removed by restriction, or handled by extending V as a partial isometry. Theorem 4.5 (Representation Invariance). Let R and R ′ realize the same operational pair. For every outcome i and coarse event A⊆{1,...,n}, T R (i) = T R ′ (i), T R (A) = T R ′ (A).(12) 5

Gauge Freedom and the Born RuleMichael Tolbert Proof. Section 3 constructs abundance before using the Born rule. In realization R the represen- tation theorem gives T R (A) =∥Q R A Ψ R ∥ 2 , Q R A := X i∈A Q R i .(13) By Lemma 4.4, V Q R A = Q R ′ A V , hence ∥Q R ′ A Ψ R ′ ∥ 2 =∥Q R ′ A V Ψ R ∥ 2 (14) =∥V Q R A Ψ R ∥ 2 (15) =∥Q R A Ψ R ∥ 2 .(16) Thus T R ′ (A) = T R (A). In nonminimal spaces, all relevant vectors lie in the initial generated subspace on which the partial isometry is norm preserving. Corollary 4.6 (Descent to the operational equivalence class). For each finite-dimensional opera- tional pair (ρ,E), the induced functional T (ρ,E) (A) := T R (A)(17) is well defined, independent of the realization R, and satisfies T (ρ,E) (A) =∥Q R A Ψ R ∥ 2 = Tr(ρE A ).(18) 4.2 Nonminimal realizations, preparation independence, and gauge interpreta- tion A nonminimal dilation may contain ancillary sectors never reached from the embedded state. Re- striction to the generated cyclic subspace isolates the minimal core. The purification proof also resolves ensemble ambiguity: the construction acts on a purification and does not count vectors in a chosen ensemble. At the present finite-dimensional stage, gauge freedom denotes algebraic and representational re- dundancy. The operational pair (ρ,E) is invariant; purifications, Naimark spaces, ancillary bases, and uniform refinements are representatives. The paper does not claim a local gauge field, connec- tion, curvature operator, or associated dynamics. 5 Approximate Representation Invariance 5.1 Stability under perturbations of state and effect Let (ρ,E) and (σ,F ) be operational descriptions. Trace-norm/operator-norm duality gives |T (E i | ρ)− T (F i | σ)|≤| Tr[(ρ− σ)E i ]| +| Tr[σ(E i − F i )]|(19) ≤∥ρ− σ∥ 1 +∥E i − F i ∥ ∞ .(20) With D(ρ,σ) = 1 2 ∥ρ− σ∥ 1 , |T (E i | ρ)− T (F i | σ)|≤ 2D(ρ,σ) +∥E i − F i ∥ ∞ .(21) Proposition 5.1 (Operational perturbation bound). For density operators ρ,σ and effects 0 ≤ E,F ≤ I, |T (E | ρ)− T (F | σ)|≤ 2D(ρ,σ) +∥E− F∥ ∞ .(22) 6

Gauge Freedom and the Born RuleMichael Tolbert 5.2 Robustness of gauge descent under approximate decoherence Let {|r i ⟩} n i=1 be normalized, linearly independent records with Gram matrix G ij = ⟨r i | r j ⟩ and |G ij |≤ ε for i̸= j. Then ∥G− I∥ ∞ ≤ (n− 1)ε =: δ.(23) Assume δ < 1. The parameter n counts records retained in the finite coarse event family, not environmental dimension or a microscopic census of branches. For increasing n k , the estimate remains informative when (n k − 1)ε k → 0. Step 1: Löwdin symmetric orthogonalization. Let R : C n →H S be the synthesis operator, Rc = P i c i |r i ⟩, so R † R = G. Define U := RG −1/2 , |e i ⟩ := U|i⟩ = X j (G −1/2 ) ji |r j ⟩.(24) Then U † U = I, and the polar factor yields ∥R− U∥ ∞ =∥G 1/2 − I∥ ∞ ≤ 1− √ 1− δ = δ 1 + √ 1− δ =: b(δ).(25) Consequently, ∥|r i ⟩−|e i ⟩∥≤ b(δ) [5]. Step 2: Rank-one subspace perturbation. Define R i =|r i ⟩⟨r i | and Π i =|e i ⟩⟨e i |. Then ∥R i − Π i ∥ ∞

q 1−|⟨r i | e i ⟩| 2 ≤ b(δ).(26) This is the rank-one specialization of Davis–Kahan-style subspace perturbation theory [6]. Step 3: Lipschitz integration. Proposition 5.1, with the state fixed, gives |T (R i | ρ)− T (Π i | ρ)|≤∥R i − Π i ∥ ∞ .(27) Theorem 5.2 (Finite-Overlap Approximate Invariance). Let {|r i ⟩} n i=1 be normalized, linearly in- dependent records with |⟨r i | r j ⟩|≤ ε for i̸= j, and set δ = (n− 1)ε < 1. Let {|e i ⟩} be their Löwdin orthogonalization. Then, for every density operator ρ, |T (R i | ρ)− T (Π i | ρ)|≤ b(δ), b(δ) = 1− √ 1− δ.(28) In particular, |T (R i | ρ)− Tr(ρΠ i )|≤ b((n− 1)ε),(29) and the bound tends to zero as ε→ 0 for fixed n. Proof. The Gram estimate makes G positive definite. Löwdin orthogonalization defines the isome- try U = RG −1/2 . Spectral localization places the spectrum of G in [1−δ, 1 +δ], yielding the vector bound above. The exact rank-one projector formula gives ∥R i − Π i ∥ ∞ ≤ b(δ). Proposition 5.1 completes the argument, and Theorem 3.1 gives T (Π i | ρ) = Tr(ρΠ i ). 7

Gauge Freedom and the Born RuleMichael Tolbert The result is effect-wise. A nonorthogonal family {R i } need not sum to the identity and is not automatically a POVM. A complete measurement-level theorem must control normalization, coarse graining, higher-rank sectors, and state preparation. For a positive family{F i } with frame operator S = P i F i , a future extension may analyze E i = S −1/2 F i S −1/2 (30) when S is positive and invertible. 6 Structural Consequences and Quantum-Information Relevance Uniform and common refinement remove dependence on representative number. Purification in- variance removes dependence on an ensemble or reference-system realization. Naimark invariance removes dependence on the projective realization of a POVM. Together they support a single principle: a physical branch measure must be constant under transformations that preserve the represented operational event. Naimark dilation guarantees that POVM statistics can be represented projectively once a proba- bility rule is already available. The present argument reverses the logical order: it first invokes the projective abundance construction as a pre-probabilistic counting mechanism and only afterward obtains the trace formula as the compressed representation of the derived measure. The formalism applies to quantum state and detector tomography, minimum-error and unambigu- ous state discrimination, noisy measurement models, and ancilla-assisted generalized measurement protocols. These applications share the same structural feature: effects are operational, while imple- mentations may employ different ancillary systems, couplings, purifications, or projective readouts. Theorem 4.5 ensures that the measure depends on (ρ,E), not on the selected realization. 6.1 Qubit test case: the trine POVM Consider H S = C 2 prepared in |0⟩. Define |φ 1 ⟩ =|0⟩,(31) |φ 2 ⟩ =− 1 2 |0⟩ + √ 3 2 |1⟩,(32) |φ 3 ⟩ =− 1 2 |0⟩− √ 3 2 |1⟩.(33) The effects E i

2 3 |φ i ⟩⟨φ i |, i = 1, 2, 3,(34) satisfy P i E i = I S , and for ρ =|0⟩⟨0|, Tr(ρE 1 ) = 2 3 ,Tr(ρE 2 ) = Tr(ρE 3 ) = 1 6 .(35) A canonical realization on H S ⊗ C 3 is W|χ⟩ = X i p E i |χ⟩⊗|i⟩ A , Q R i = I S ⊗|i⟩ A ⟨i| A .(36) 8

Gauge Freedom and the Born RuleMichael Tolbert The dilated state is |Ψ R ⟩ = r 2 3 |φ 1 ⟩|1⟩ A − 1 √ 6 |φ 2 ⟩|2⟩ A − 1 √ 6 |φ 3 ⟩|3⟩ A .(37) Thus the projective branch norms are 2/3, 1/6, 1/6, represented by four, one, and one descendants in a six-element homogeneous refinement. Under a unitary U A on the ancilla, |Ψ R ′ ⟩ = (I S ⊗ U A )|Ψ R ⟩, Q R ′ i = (I S ⊗ U A )Q R i (I S ⊗ U A ) † ,(38) and ∥Q R ′ i Ψ R ′ ∥ 2 =∥Q R i Ψ R ∥ 2 . 7 Principal Objections and Replies 7.1 Does purification hide a preferred ensemble? No ensemble decomposition is selected. The construction is performed on a purification, and invariance under changes of purification makes the event measure a function of the reduced density operator alone. 7.2 Does Naimark dilation assume the Born rule? Naimark dilation is used as an operator-theoretic representation of effects by projectors. The measure is constructed on the projective realization by the abundance theorem and only afterward compressed to the system. The construction is pre-probabilistic in a limited sense: it defines structural abundance over primitive-equivalent representatives before assigning an epistemic or stochastic role. 7.3 Is dilation invariance trivial once the trace formula is written? Equality of trace expressions is immediate after the Born form is known. The conceptual burden is to show that the pre-probabilistic abundance construction is compatible with unitary equivalence and ignores unused ancillary sectors. Theorem 4.5 makes that descent explicit. 7.4 Is “gauge freedom” overstated? The paper establishes a concrete gauge-like structure at the level of representation: mathematically distinct realizations induce identical event measures. It does not yet construct a local gauge field, connection, curvature operator, or dynamics. The theorem is the finite-dimensional foundation of that larger program. 7.5 Does the framework solve approximate decoherence? Theorem 5.2 proves a finite-dimensional, effect-wise robustness result for approximately orthogonal normalized records. Extending the result to complete normalized POVMs, higher-rank sectors, continuous outcomes, and geometric transport remains open. 9

Gauge Freedom and the Born RuleMichael Tolbert 7.6 Research program and future directions The research program is cumulative: Paper 1 develops finite pure-state PVMs; the present paper treats mixed states and finite POVMs; anticipated Paper 3 addresses complete measurement archi- tectures and continuous realization families; and anticipated Paper 4 seeks operator-algebraic and Type III extensions. The later stages are not assumed by the present theorems. 8 Conclusion The unitary-refinement program extends from finite pure-state PVMs to finite mixed-state POVMs by combining purification, Naimark dilation, and the original cardinality representation theorem. The enlarged-space construction first fixes cardinal abundance and then induces an event measure represented by ∥Q i Ψ∥ 2 . Compression yields Tr(ρE i ). The central advance is Representation In- variance: equivalent purification-dilation realizations generate the same induced measure, so the measure descends to the operational pair. The finite-overlap theorem adds an explicit effect-wise robustness result. For fixed record number n, approximately orthogonal rank-one records induce typicalities converging to ideal projective values as ε→ 0. This supplies a rigorous bridge from microscopic overlap to controlled measure error and provides a foundation for later geometric and operator-algebraic extensions. 10

Gauge Freedom and the Born RuleMichael Tolbert A Proof Architecture at a Glance Operational input: density operator rho and finite POVM {E_i} Unified realization: purification and Naimark dilation to |Psi> and {Q_i} Projective abundance: equal-norm refinement and cardinal composition Generalized Born form: T_R(i) = ||Q_i Psi||^2 = Tr(rho E_i) Representation descent: invariance across equivalent realizations Approximate extension: controlled changes for nearby effects and records 11

Gauge Freedom and the Born RuleMichael Tolbert References [1] Michael Tolbert. Typicality, unitary refinement, and the origin of the born rule. Manuscript, 2026. [2] Lane P. Hughston, Richard Jozsa, and William K. Wootters. A complete classification of quantum ensembles having a given density matrix. Physics Letters A, 183:14–18, 1993. [3] M. A. Naimark. On a representation of additive operator set functions. C. R. (Doklady) Acad. Sci. URSS (N.S.), 41:359–361, 1943. [4] Roberto Beneduci. Notes on naimark’s dilation theorem. Journal of Physics: Conference Series, 1638:012006, 2020. [5] Per-Olov Löwdin. On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals. Journal of Chemical Physics, 18:365–375, 1950. [6] Chandler Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. iii. SIAM Journal on Numerical Analysis, 7(1):1–46, 1970. 12