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From Quantum Measurement to Gauge Structure:
Minimal Response and Absolute Canonicity
Michael Tolbert
Paper 3 Version 2.0
Abstract
Generalized quantum measurements admit many purification and dilation realizations of the
same operational state–measurement data. On each finite-dimensional regular fixed-rank stratum,
this representation freedom forms a natural gauge structure: framed minimal purification and
Naimark data constitute a smooth principal bundle whose compact fibers are exactly the auxiliary
gauge orbits. Complete-POVM robustness controls perturbations of states, labeled measurement
architectures, fixed coarse events, canonical normalization, and smooth paths.
A right-invariant framed Hilbert–Schmidt response geometry assigns every operational tan-
gent a unique minimum-response lift. These lifts define a smooth gauge-equivariant principal
connectionH
fr
and a positive gauge-independent response metric on the operational base.
The geometric construction establishes transport and relative canonicity independently of any
informational identity premise.
To address physical uniqueness, the paper introduces Quantum Informational Identity (QII)
and Dynamical Quantum Informational Identity as explicit foundational assumptions. The
generated carrier, labeled event algebra, and admitted covariance group are defined explicitly
from the fixed framed architecture. Exact gauge intertwiners preserve this profile pointwise,
and smooth gauge paths preserve its dynamical history. Under Dynamical QII and an explicit
profile-closure restriction, all profile-closed admissible principal connections determine the same
represented transport content. Relative to the framed Hilbert–Schmidt response geometry,H
fr
is the unique minimum-response representative. This scoped conclusion is called profile-relative
absolute canonicity; it does not claim that informational identity derives the Hilbert–Schmidt
response geometry or excludes extensions with additional observables.
Version status. Version 2.0 incorporates the peer-review revision of the informational canonicity
layer. The generated carrier and admitted covariance group are now defined explicitly from the fixed
framed architecture, gauge covariance is proved rather than asserted, pointwise QII is separated from
Dynamical QII, and the final uniqueness claim is stated relative to profile closure. The geometric
and response results of Sections 2–6 are retained.
1 Introduction
Quantum measurements are operationally specified by states, labeled effects, and their event
assignments, yet the purification and dilation data used to represent those measurements are not
unique. This raises two questions: how is the freedom among realization descriptions organized
geometrically, and is there a distinguished way to transport realizations along changing operational
data?
The central claim is that realization freedom forms a natural gauge structure and that its framed
geometry selects a unique minimum-response transport. On each regular fixed-rank state–POVM
stratum, minimal purification amplitudes and framed Naimark realizations assemble into a smooth
1
principal bundle. Its fibers are exactly the orbits of a compact auxiliary gauge group. Gauge
structure therefore arises from the multiplicity of realization descriptions of the same operational
measurement data.
A right-invariant Hilbert–Schmidt metric separates vertical gauge motion from changes required
by an operational tangent. Orthogonal projection selects the unique lift of least framed norm. The
resulting horizontal distributionH
fr
is a smooth principal connection, and its squared horizontal
norm descends to a positive response metric on the operational base. This construction is related
in spirit to purification-bundle approaches to mixed-state parallel transport, while extending the
realization space to include labeled Naimark data and complete measurement architectures [1,2,3].
The geometric result establishes relative canonicity. To address represented physical uniqueness,
the paper defines a fixed generated carrier and a label-preserving covariance group, then introduces
pointwise QII and Dynamical QII. Dynamical QII yields transport-content identity only within
the explicitly profile-closed class of admissible connections. The framed Hilbert–Schmidt geometry
selectsH
fr
as the unique minimum-response representative. The manuscript calls this scoped
two-part result profile-relative absolute canonicity.
1.1 Roadmap of the argument
The proof architecture is theorem-driven and separates construction from interpretation:
1.Operational domain. Regular fixed-rank state and labeled-POVM strata provide a smooth
base with fixed auxiliary dimensions and stable event correspondence.
2.
Gauge bundle. Framed purification–dilation data form a smooth principal bundle; each
fiber is exactly one compact gauge orbit, and the vertical tangent space is the infinitesimal
gauge action.
3.Robustness. Complete-POVM estimates control states, all labeled effects, fixed coarse
grainings, canonical normalization, and smooth operational paths.
4.Minimal response. The framed Hilbert–Schmidt metric selects a unique minimum-norm
lift of each operational tangent, producing the principal connectionH
fr
and the operational
response metric.
5.
Informational identity. Exact gauge transformations preserve the generated informational
profile; smooth gauge paths preserve the corresponding dynamical profile.
6.Final theorem. Dynamical QII and explicit profile closure yield unique represented transport
content, while the framed response geometry supplies its unique minimum-response represen-
tative. These conclusions constitute profile-relative absolute canonicity in the scoped sense
adopted here.
The paper remains finite dimensional and restricted to regular fixed-rank strata, fixed outcome
labels, minimal generated realizations, and the represented informational structures defined below.
Rank-changing boundaries, nonregular POVMs, continuous outcomes, and a possible uniqueness
derivation of the response geometry itself remain outside the present theorem.
2
2 Operational Parameter Space
2.1 Ambient operator spaces
LetH
S
be a complex Hilbert space of finite dimensiond. LetHerm(H
S
) be the real vector space of
Hermitian operators on H
S
, equipped initially with the Hilbert–Schmidt inner product
⟨X,Y⟩
HS
= Tr(XY ).
Fix a finite labeled outcome set
I ={1,...,n}.
The labels are part of the operational specification and remain fixed along every admissible path
considered in the main theorems.
Definition 2.1 (Fixed-rank state stratum). For 1≤ r ≤ d, define
D
r
(H
S
) :={ρ∈ Herm(H
S
) : ρ≥ 0, Trρ = 1, rankρ = r}.
The support projection ofρis denotedp
ρ
. The full-rank case isr=d, but lower fixed ranks are
retained because the minimal purification dimension is then constant on the stratum.
Proposition 2.2 (Smooth state stratum). The fixed-rank density-operator setD
r
(H
S
) is a smooth
real manifold of dimension
dimD
r
(H
S
) = 2dr− r
2
− 1.
Its tangent space at ρ is naturally identified with
T
ρ
D
r
n
X = X
†
: (I− p
ρ
)X(I− p
ρ
) = 0, TrX = 0
o
.
Proof sketch.
Writeρ=AA
†
, whereA:C
r
→H
S
has full column rank andTr(AA
†
) = 1. The unit
sphere of full-column-rank amplitudes has real dimension 2dr−1. The free right actionA7→ AUof
U(r) has dimensionr
2
and leavesAA
†
unchanged. The quotient therefore has dimension 2dr−r
2
−1;
see, for example, standard quotient models of fixed-rank positive-semidefinite manifolds [4,5,6].
Differentiating a smooth rank-preserving curve gives the displayed support-block and trace conditions.
Conversely, writeH
S
=ranp
ρ
⊕ kerρand decompose an admissibleXasX=
X
11
X
12
X
†
12
0
. Letρ
+
be the positive-definite restriction ofρtoranp
ρ
, setK:=X
†
12
ρ
−1
+
, and choose the skew-adjoint
block operatorA=
0 −K
†
K0
. For smallt, defineρ(t) :=e
tA
ρ+t(X
11
−[A,ρ]
11
)
e
−tA
, with the
middle operator acting on the moving rank-rsupport and extended by zero. Positivity and rankr
persist for sufficiently smallt,Trρ(t) = 1 becauseTrX= 0, and direct differentiation att= 0 gives
̇ρ(0) = X. Thus every displayed tangent direction is realized by a rank-preserving curve.
2.2 Fixed-rank POVM strata
Choose a rank pattern
r = (r
1
,...,r
n
),1≤ r
i
≤ d.
LetPos
r
i
(H
S
) denote the smooth manifold of positive semidefinite Hermitian operators of rankr
i
.
Its real dimension is 2dr
i
− r
2
i
.
Consider the smooth summation map
Σ :
n
Y
i=1
Pos
r
i
(H
S
)−→ Herm(H
S
),Σ(E
1
,...,E
n
) =
n
X
i=1
E
i
.
3
The rank pattern alone does not guarantee that the level set Σ
−1
(I) is smooth at every point. The
manuscript therefore works on the regular locus of the summation constraint.
Definition 2.3 (Fixed-rank POVM locus). Define
POVM
r
(H
S
) :=
(
(E
1
,...,E
n
) : E
i
≥ 0, rank(E
i
) = r
i
,
X
i
E
i
= I
)
.
This locus need not be smooth at every point.
Assumption 2.4 (Admissible nonempty rank pattern). Throughout the remainder of the paper,
fix a rank patternr for which
POVM
reg
r
(H
S
)̸= ∅.
In particular,
P
i
r
i
≥ d
is necessary because the ranges of the effects must collectively spanH
S
when
P
i
E
i
= I.
Definition 2.5 (Regular fixed-rank POVM stratum). Define
POVM
reg
r
(H
S
) :=
(
(E
1
,...,E
n
) : E
i
≥ 0, rank(E
i
) = r
i
,
X
i
E
i
= I, dΣ
E
is surjective
)
.
Here
dΣ
E
(X
1
,...,X
n
) =
X
i
X
i
.
Proposition 2.6 (Smooth POVM stratum). The regular fixed-rank POVM setPOVM
reg
r
(H
S
) is a
smooth embedded submanifold of
Q
i
Pos
r
i
(H
S
). Its dimension is
dim POVM
reg
r
(H
S
) =
n
X
i=1
(2dr
i
− r
2
i
)− d
2
.
At E = (E
i
), its tangent space is
T
E
POVM
reg
r
(
(X
i
)
n
i=1
: X
i
= X
†
i
, (I− p
i
)X
i
(I− p
i
) = 0,
X
i
X
i
= 0
)
,
where p
i
is the support projection of E
i
.
Proof.
By definition,Iis a regular value of the restricted summation map at every point of
POVM
reg
r
(H
S
). The regular-value theorem gives a smooth embedded level set of codimension
dim Herm(H
S
) =d
2
[7]. The tangent space is the kernel ofdΣ
E
inside the product of the fixed-rank
positive tangent spaces.
2.3 The regular operational stratum
Definition 2.7 (Regular operational stratum). For fixed d,n,r, andr, define
O
reg
(r,r) :=D
r
(H
S
)× POVM
reg
r
(H
S
).
An operational point is written o = (ρ,E), where E = (E
1
,...,E
n
).
4
Its dimension is
dimO
reg
= (2dr− r
2
− 1) +
X
i
(2dr
i
− r
2
i
)− d
2
.
A convenient initial Riemannian metric is the product Hilbert–Schmidt metric
∥(δρ,δE)∥
2
op
=∥δρ∥
2
HS
+
X
i
∥δE
i
∥
2
HS
.
The later robustness theory may supplement this smooth metric with trace norm for states and
operator norm for effects. The smooth geometry and the perturbation metric need not coincide, but
both must be stated explicitly.
2.4 Label preservation and event correspondence
Assumption 2.8 (Label-Preservation Hypothesis). Every operational path is a map
t7−→ o(t) =
ρ(t),E
1
(t),...,E
n
(t)
∈O
reg
with one fixed labeled index setI. No discontinuous permutation or relabeling of outcomes is
allowed inside an admissible chart. If a relabeling is physically intended, it must be represented by
an explicit permutation action and carried through every event identification.
This assumption separates smooth transport from an independent labeling convention. It
prevents a path from silently identifying one outcome sector with an unrelated sector merely because
both occupy the same numerical position in a list.
Definition 2.9 (Fine-event correspondence). For every subset A⊆ I and every t, define
E
A
(t) :=
X
i∈A
E
i
(t).
The same label subset A is used at every point of the path.
Definition 2.10 (Fixed coarse-graining correspondence). Let
Γ ={B
1
,...,B
m
}
be a fixed partition of I. The associated coarse measurement along the path is
E
Γ
α
(t) :=
X
i∈B
α
E
i
(t), α = 1,...,m.
A coarse event C ⊆{1,...,m} corresponds to the fixed fine-label union
S
α∈C
B
α
.
The main robustness and transport theorems treat Γ as fixed. Time-dependent merging or
splitting of outcome blocks is a separate problem and is not included in the minimal paper.
Proposition 2.11 (Smooth event correspondence). Ift 7→ E
i
(t) isC
k
for every labeli, then
t7→ E
A
(t) and t7→ E
Γ
α
(t) are C
k
for every fixed fine event A and every fixed coarse block B
α
.
Proof. Finite summation is linear and therefore smooth.
5
2.5 Boundaries and scope
The regular stratum excludes points where the rank ofρor an effect changes, where the regular-value
condition fails, or where the chosen outcome correspondence ceases to be meaningful. Such points
may require a stratified-space treatment. No global smoothness claim is made across those loci.
2.6 A two-outcome qubit example
A simple example illustrates what is operational and what is gauge. LetH
S
=C
2
, choose a full-rank
state
ρ =
a0
0 1− a
!
,0 < a < 1,
and a labeled two-outcome POVM
E
1
p 0
0 q
!
, E
2
= I− E
1
,0 < p,q < 1.
Both effects are full rank, and the pair lies in the regular full-rank stratum. One framed realization
is obtained from
X = ρ
1/2
, W
1
= E
1/2
1
, W
2
= E
1/2
2
,
with W = (W
1
,W
2
)
T
. For arbitrary unitaries U,V
1
,V
2
∈ U (2),
(X,W
1
,W
2
)7−→ (XU,V
†
1
W
1
,V
†
2
W
2
)
changes the auxiliary frames but leavesρ,E
1
, andE
2
unchanged. These descriptions therefore
occupy one gauge fiber.
Now leta,p, andqvary smoothly with time while remaining inside (0,1). The operational
path changes the state and measurement effects, while a realization lift must also choose how
the auxiliary frames vary. Adding any smooth time-dependentU(t),V
1
(t),V
2
(t) changes that lift
without changing the operational path. The vertical part of this change is gauge motion. The
framed minimum-response connection removes the gratuitous vertical component relative tog
fr
and
selects one horizontal lift. QII does not derive that metric; it identifies gauge-related lifted histories
as carrying the same represented transport content. This example displays the two logical tasks of
the final theorem: informational identity fixes the content class, and minimum response selects its
framed representative.
3 Purification–Dilation Realization Space
3.1 Fixed model auxiliary spaces
On O
reg
(r,r), the supported auxiliary dimensions are constant. Fix the model purification space
A := C
r
.
Fix the model generated Naimark space
K :=
n
M
i=1
C
r
i
, m := dimK =
X
i
r
i
,
with canonical coordinate projections P
i
onto the ith block.
6
3.2 Framed purification and dilation data
The local construction is simplest when auxiliary realizations are represented by amplitudes and
frames rather than by varying Hilbert spaces.
Definition 3.1 (Purification amplitude). A minimal purification amplitude is a full-column-rank
map
X :A−→H
S
satisfying
XX
†
= ρ,Tr(XX
†
) = 1.
The choice is not unique: X and XU represent the same state for every U ∈ U (r).
Definition 3.2 (Amplitude–vector identification). Fix an orthonormal reference basis{|a⟩}
r
a=1
of
A = C
r
. For an amplitude X :A→H
S
, define
|φ
X
⟩ :=
r
X
a=1
X|a⟩⊗|a⟩∈H
S
⊗A.
Then
Tr
A
|φ
X
⟩⟨φ
X
| = XX
†
.
For U ∈ U (r),
|φ
XU
⟩ = (I
H
S
⊗ U
T
)|φ
X
⟩,
where the transpose is taken in the fixed reference basis. This convention fixes the purifying-factor
gauge transformation used below.
Definition 3.3 (Framed minimal Naimark realization). ForE= (E
i
) withrank(E
i
) =r
i
, a framed
minimal Naimark realization is a block isometry
W =
W
1
.
.
.
W
n
:H
S
−→K
such that
W
†
i
W
i
= E
i
,
X
i
W
†
i
W
i
= I.
EachW
i
:H
S
→ C
r
i
has full row rank. The construction is the finite-dimensional labeled form of a
minimal Naimark dilation [8, 9]. The fixed coordinate projectors satisfy
W
†
P
i
W = E
i
.
3.3 The framed realization space
Definition 3.4 (Framed Purification–Dilation Realization Space). LetA
r
be the normalized
full-column-rank amplitude space
A
r
:=
n
X ∈ Hom(A,H
S
) : rankX = r, Tr(XX
†
) = 1
o
,
and let W
r
be the block-rank Stiefel stratum
W
r
:=
(
(W
1
,...,W
n
) : rankW
i
= r
i
,
X
i
W
†
i
W
i
= I
)
.
7
Define the ambient compression map
π
amb
:A
r
×W
r
−→D
r
(H
S
)× POVM
r
(H
S
)
by
π
amb
(X,W ) :=
XX
†
,{W
†
i
W
i
}
n
i=1
.
The framed Purification–Dilation Realization Space over the regular operational stratum is
R
reg
:= π
−1
amb
(O
reg
).
The realization projection is the restriction
π := π
amb
R
reg
:R
reg
−→O
reg
.
For o = (ρ,E)∈O
reg
, the fiber is
F
o
:= π
−1
(o).
Remark 3.5 (Why the regular-image condition is explicit). Fixed block ranks and the normalization
P
i
W
†
i
W
i
=Ido not by themselves guarantee that the compressed POVM belongs to the regular-
value locus of Definition 2.5. The preimage condition in Definition 3.4 makes the codomain ofπ
well defined and restricts the realization geometry to the same regular stratum used throughout the
paper.
This framed realization space provides a fixed finite-dimensional ambient model for the supported
realization data of Paper 2.
3.4 Unified state and generated projectors
GivenR= (X,W) overo= (ρ,E), let|φ
X
⟩∈H
S
⊗Abe the purification vector associated withX.
Define
|Ψ
R
⟩ := (W ⊗ I
A
)|φ
X
⟩, Q
i
:= P
i
⊗ I
A
.
For every label event A⊆ I, set
Q
A
:=
X
i∈A
Q
i
.
The inherited refinement-derived typicality is represented by
T
R
(A) :=∥Q
A
Ψ
R
∥
2
.
Paper 2 proves that this value descends to the operational pair and is represented byTr(ρE
A
) only
after the projective abundance construction has been applied. Paper 3 retains that logical order.
3.5 Exact gauge action
Let
G := U (r)×
n
Y
i=1
U (r
i
).
For g = (U,V
1
,...,V
n
), define the right action
(X,W
1
,...,W
n
)· g := (XU,V
†
1
W
1
,...,V
†
n
W
n
).
8
This is a genuine right action: for g,h∈ G,
(R· g)· h = R· (gh).
It changes the auxiliary purification frame and the block outcome frames while leaving the compressed
operational pair unchanged. The adjoints on the Naimark-frame factors reconcile their natural
left unitary change of coordinates with the right-action convention used for principal realization
geometry.
Proposition 3.6 (Gauge invariance of the projection). For every R∈R
reg
and g ∈ G,
π(R· g) = π(R).
Proof. Right multiplication gives
(XU )(XU )
†
= XUU
†
X
†
= XX
†
.
For every outcome block,
(V
†
i
W
i
)
†
(V
†
i
W
i
) = W
†
i
V
i
V
†
i
W
i
= W
†
i
W
i
.
Thus the state and every labeled effect are unchanged. Because regularity is a property of the
compressed operational point, the action also preserves R
reg
.
Proposition 3.7 (Fibers are exact gauge orbits). LetR= (X,W) andR
′
= (X
′
,W
′
) lie inR
reg
.
Then
π(R) = π(R
′
)
if and only if there exists g ∈ G such that
R
′
= R· g.
Proof.
One direction follows from Proposition 3.6. Conversely,XX
†
=X
′
X
′†
withX,X
′
of full
column rank implies X
′
= XU for some U ∈ U (r). For each i, the identities
W
†
i
W
i
= W
′†
i
W
′
i
and the full row ranks imply, by polar decomposition on the common support, thatW
′
i
=L
i
W
i
for
some L
i
∈ U (r
i
). Set V
i
:= L
†
i
. Then W
′
i
= V
†
i
W
i
, so
R
′
= R· (U,V
1
,...,V
n
).
3.6 Freeness, stabilizers, and quotient
Proposition 3.8 (Free framed gauge action). The action of G on R
reg
is free.
Proof.IfXU=XandXhas full column rank, thenU=I
r
. IfV
†
i
W
i
=W
i
andW
i
has full
row rank, thenV
†
i
=I
r
i
and henceV
i
=I
r
i
. Therefore only the identity element fixes a framed
realization.
BecauseGis compact, its smooth action is proper [7]. Proposition 3.7 identifies the orbit space
R
reg
/Gset-theoretically withO
reg
. A smooth principal-bundle conclusion is postponed until the
next stage verifies the smooth structure ofR
reg
, the submersion property ofπ, and smooth local
sections.
9
3.7 Exact gauge invariance of typicality
Proposition 3.9 (Vertical invariance of refinement-derived typicality). IfR
′
=R· glies in the
same exact gauge orbit and A⊆ I is a fixed label event, then
T
R
′
(A) = T
R
(A).
Proof. By Definition 3.2, the purification-frame change X 7→ XU is implemented on the purifying
factor by U
T
. The block-frame changes W
i
7→ V
†
i
W
i
combine into the block-diagonal unitary
V
K
:=
n
M
i=1
V
†
i
onK. BecauseV
K
commutes with every coordinate projectorP
i
, the unitaryV
K
⊗ U
T
onK⊗A
maps|Ψ
R
⟩to|Ψ
R
′
⟩and intertwines every labeledQ
i
=P
i
⊗ I
A
, hence everyQ
A
. Unitary norm
preservation gives
∥Q
A
Ψ
R
∥ =∥Q
A
Ψ
R
′
∥,
which proves the claim. This is the realization-space form of the Representation Invariance Theorem
of Paper 2.
Remark 3.10 (Infinitesimal gauge directions). For the corrected right action, a Lie-algebra element
(ξ,η
1
,...,η
n
)∈ u(r)⊕
n
M
i=1
u(r
i
)
generates the infinitesimal variation
δX = Xξ, δW
i
=−η
i
W
i
.
These directions are tangent to the exact gauge orbit and will be identified with the vertical tangent
space after the local realization geometry establishes that the realization projection is a smooth
submersion.
3.8 Lifted event correspondence
Definition 3.11 (Lifted event correspondence). For a label-preserving operational patho(t) and a
realization lift R(t), the event A⊆ I is represented in the fixed framed model by
Q
A
:=
X
i∈A
Q
i
, Q
i
= P
i
⊗ I
A
.
For a fixed coarse partition Γ ={B
α
}, define
Q
Γ
α
:=
X
i∈B
α
Q
i
.
These coordinate projectors are independent oftbecauseK,A, and the labeled coordinate projections
P
i
are fixed in the framed realization space.
Thus event correspondence is carried by the fixed labels, not by motion of the coordinate
projectors. The nontrivial transport problem concerns the amplitudesX(t), the Naimark isometries
W(t), the unified states
Ψ
R(t)
E
, and the gauge frames used to compare neighboring realizations.
Under changes of framed chart, transition functions preserve the labeled projector decomposition.
10
Assumption 3.12 (Event-Compatible Lift). An admissible realization liftt7→ R(t) must project
too(t) and preserve the fixed outcome labels and coarse partition. In the fixed framed model,
the coordinate event projectorsQ
A
remain constant. Gauge transformations along the lift must
commute with the labeled coordinate projectors unless an explicit permutation action is included.
Horizontal transport will therefore compareX(t),W(t), and
Ψ
R(t)
E
relative to this fixed event
correspondence.
3.9 Interim structural summary
The preceding construction establishes:
•
a conservative smooth operational domainO
reg
defined by fixed ranks and a regular-value
condition;
• explicit label-preserving correspondence for fine and coarse events;
• a fixed-dimensional framed Purification–Dilation Realization Space R
reg
;
• a projection π whose fibers are exactly the orbits of the compact gauge group G;
• exact vertical invariance of the refinement-derived typicality;
•
a clearly stated transition from exact fiber-orbit structure to the local principal-bundle theorem
proved in Section 4.
4 Local Realization Geometry
This section proves the local geometric structure required for the realization bundle. All statements
are restricted to one regular fixed-rank operational stratumO
reg
(r,r) and to the framed realization
space R
reg
of Definition 3.4. No assertion is made across rank-changing or nonregular loci.
4.1 Smooth structure of the framed realization space
Define the normalized amplitude manifold
A
r
n
X ∈ Hom(A,H
S
) : rankX = r, Tr(XX
†
) = 1
o
and the block-rank Stiefel stratum
W
r
n
W = (W
1
,...,W
n
) : W
†
W = I, rankW
i
= r
i
o
.
Recall that R
reg
is the preimage of O
reg
under the ambient compression map π
amb
.
Proposition 4.1 (Smooth framed realization manifold). The spaceR
reg
is a finite-dimensional
smooth manifold. Its dimension is
dimR
reg
= (2dr− 1) + (2md− d
2
), m =
n
X
i=1
r
i
.
Equivalently,
dimR
reg
= dimO
reg
n
X
i=1
r
2
i
.
11
Proof.The full-column-rank condition onXis open inHom(A,H
S
). On that open set, the smooth
function
F (X) = Tr(XX
†
)
has differential
dF
X
(
̇
X) = 2 Re Tr(X
†
̇
X).
BecausedF
X
(X) = 2Tr(X
†
X)>0, the value 1 is regular. HenceA
r
=F
−1
(1) is a smooth
hypersurface of real dimension 2dr− 1.
Writing the blocks as one operator
W :H
S
−→K,
the conditionW
†
W=Idefines the complex Stiefel manifold of isometric embeddings ofH
S
into
K, of real dimension 2md− d
2
[10,5]. The conditionsrankW
i
=r
i
are open, soW
r
is an open
submanifold of that Stiefel manifold.
Surjectivity of a linear map between finite-dimensional vector spaces is an open condition. Hence
the regular-value condition definingPOVM
reg
r
(H
S
) is open in the fixed-rank POVM locus. Therefore
O
reg
is open in D
r
(H
S
)× POVM
r
(H
S
), and
R
reg
= π
−1
amb
(O
reg
)
is open in A
r
×W
r
. This proves smoothness and the first dimension formula. For the second,
dimO
reg
2dr− r
2
− 1 +
X
i
(2dr
i
− r
2
i
)− d
2
!
+
r
2
+
X
i
r
2
i
!
= 2dr− 1 + 2d
X
i
r
i
− d
2
= (2dr− 1) + (2md− d
2
),
where m =
P
i
r
i
.
4.2 The realization projection
Proposition 4.2 (Smooth surjective realization projection). The realization projection
π :R
reg
−→O
reg
is smooth and surjective. Its differential at R = (X,W ) is
dπ
R
(
̇
X,
̇
W ) =
̇
XX
†
- X
̇
X
†
,
n
̇
W
†
i
W
i
- W
†
i
̇
W
i
o
n
i=1
.
Proof.
The mapπis the restriction of a polynomial map in the real and imaginary matrix entries,
so it is smooth, and direct differentiation gives the displayed formula.
Leto= (ρ,E)∈O
reg
. Choose a full-column-rank amplitudeX:C
r
→H
S
withXX
†
=ρ. For
each i, choose an isometry
J
i
: C
r
i
−→ suppE
i
onto the support of E
i
, and define
W
i
:= J
†
i
E
1/2
i
.
12
Then
W
†
i
W
i
= E
1/2
i
J
i
J
†
i
E
1/2
i
= E
i
,
because J
i
J
†
i
is the support projection of E
i
. Consequently,
X
i
W
†
i
W
i
X
i
E
i
= I,
so the stacked operatorWis an isometry. SinceE
1/2
i
has rankr
i
andJ
†
i
restricts to an isomorphism
fromsuppE
i
ontoC
r
i
, one hasrankW
i
=r
i
. The compressed operational point iso∈O
reg
. Thus
(X,W )∈R
reg
and π(X,W ) = o.
4.3 Smooth local realization sections
Lemma 4.3 (Local smooth support frames). LetA
0
≥0 have rankk. There is a neighborhoodU
of A
0
in the rank-k positive-semidefinite stratum and a smooth map
J : U −→ Hom(C
k
,H
S
)
such that J (A)
†
J (A) = I
k
and J (A)J (A)
†
is the support projection of A for every A∈ U.
Proof.
Letp(A) denote the support projection ofA. On a sufficiently small neighborhood ofA
0
in the rank-kpositive-semidefinite stratum, the nonzero spectrum remains separated from zero.
Choose a positively oriented contourγenclosing the nonzero spectrum ofA
0
and excluding zero.
After shrinking the neighborhood if necessary, the support projection is the Riesz projection
p(A) =
1
2πi
Z
γ
(zI− A)
−1
dz,
and therefore depends smoothly on A by smooth spectral functional calculus [11].
Fix an isometry
J
0
: C
k
−→ suppA
0
.
Define
B(A) := p(A)J
0
.
At A = A
0
, one has B(A
0
) = J
0
, so
B(A
0
)
†
B(A
0
) = I
k
.
By continuity, B(A)
†
B(A) remains positive definite on a smaller neighborhood. Set
J (A) := B(A)
B(A)
†
B(A)
−1/2
.
The inverse square-root map is smooth on the positive-definite cone, henceJ(A) is smooth. Direct
calculation gives
J (A)
†
J (A) = I
k
.
Moreover,ranB(A)⊆ ranp(A) and both spaces have dimensionk, soranJ(A) =ranp(A). There-
fore
J (A)J (A)
†
= p(A).
13
Lemma 4.4 (Smooth fixed-rank square-root factors). On the neighborhoodUof Lemma 4.3, the
maps
A7−→ A
1/2
J (A)and A7−→ J (A)
†
A
1/2
are smooth.
Proof. Write
H(A) := J (A)
†
AJ (A).
This is a smooth family of positive-definitek× kmatrices. Its spectrum stays in a compact subset
of (0,∞) after shrinkingU. The positive square-root map is smooth on positive-definite matrices,
so H(A)
1/2
varies smoothly. Because ranJ (A) = suppA, the identity
A = J (A)H(A)J (A)
†
holds on the full Hilbert space. Consequently,
A
1/2
J (A) = J (A)H(A)
1/2
and
J (A)
†
A
1/2
= H(A)
1/2
J (A)
†
.
Both maps are therefore smooth.
Theorem 4.5 (Smooth local realization sections). For everyo
0
= (ρ
0
,E
0
)∈O
reg
, there is an open
neighborhood U ⊆O
reg
of o
0
and a smooth map
s : U −→R
reg
such that
π◦ s = id
U
.
Proof.Apply Lemma 4.3 toρ
0
and to every effectE
0
i
. After shrinking to a product neighborhood
U, obtain smooth support isometries
J
ρ
(ρ) : C
r
→ suppρ
and
J
i
(E
i
) : C
r
i
→ suppE
i
.
Define
X(ρ) := ρ
1/2
J
ρ
(ρ)
and
W
i
(E) := J
i
(E
i
)
†
E
1/2
i
.
Lemma 4.4 gives smoothness. Moreover,
X(ρ)X(ρ)
†
= ρ
1/2
J
ρ
(ρ)J
ρ
(ρ)
†
ρ
1/2
= ρ
1/2
p
ρ
ρ
1/2
= ρ,
and
W
i
(E)
†
W
i
(E) = E
i
.
14
SinceJ
ρ
(ρ) is an isomorphism ontosuppρ,X(ρ) has rankr. Similarly,J
i
(E
i
)
†
is an isomorphism
fromsuppE
i
ontoC
r
i
, soW
i
(E) has rankr
i
. Because
P
i
E
i
=I, the stackedW(E) is an isometry.
Hence
s(ρ,E) :=
X(ρ),W
1
(E),...,W
n
(E)
lies in R
reg
and satisfies π(s(ρ,E)) = (ρ,E).
Corollary 4.6 (Submersion property). The realization projectionπ:R
reg
→ O
reg
is a smooth
surjective submersion.
Proof. At every o∈ U, differentiation of
π◦ s = id
U
gives
dπ
s(o)
◦ ds
o
= id
T
o
O
reg
,
so dπ
s(o)
is surjective. Every R∈ π
−1
(o) has the form R = s(o)· g by Proposition 3.7. Let
R
g
:R
reg
−→R
reg
, R
g
(R
′
) = R
′
· g.
Because π◦ R
g
= π, differentiation gives
dπ
R
′
·g
◦ d(R
g
)
R
′
= dπ
R
′
.
The mapR
g
is a diffeomorphism with inverseR
g
−1
, sod(R
g
)
R
′
is invertible. Hencedπ
R
′
·g
anddπ
R
′
have the same rank. Taking R
′
= s(o) shows that dπ
R
is surjective at every R.
4.4 Local principal realization geometry
Theorem 4.7 (Local principal realization geometry). Leto
0
∈O
reg
. There is an open neighborhood
U of o
0
and a G-equivariant diffeomorphism
Φ
s
: U × G−→ π
−1
(U ),Φ
s
(o,g) = s(o)· g,
where the right action on U × G is
(o,g)· h := (o,gh).
The trivialization satisfies
π◦ Φ
s
= pr
U
.
Consequently,
π :R
reg
−→O
reg
is a smooth principalG-bundle on each regular fixed-rank stratum in the standard local-triviality
sense [12, 7].
Proof.Letsbe the local section from Theorem 4.5. Smoothness of Φ
s
follows from smoothness ofs
and of the right action. Proposition 3.7 gives surjectivity, and freeness from Proposition 3.8 gives
injectivity.
It remains to prove that the inverse is smooth. Let
s(o) =
X
s
(o),W
s,1
(o),...,W
s,n
(o)
15
and let
R = (X,W
1
,...,W
n
)∈ π
−1
(U ), o = π(R).
The matricesX
s
(o) have constant full column rankr, and the matricesW
s,i
(o) have constant full
row rank r
i
. Moore–Penrose inversion is smooth on a fixed-rank matrix manifold [13]. Define
U
R
:= X
s
(o)
+
X
and
V
i,R
:=
W
i
W
s,i
(o)
+
†
.
By Proposition 3.7, there is a unique unitary gauge elementg= (U,V
1
,...,V
n
) such thatR=s(o)·g.
Multiplying the identities
X = X
s
(o)U
and
W
i
= V
†
i
W
s,i
(o)
by the corresponding one-sided pseudoinverses shows that the recovered matrices above equal those
unitary gauge coordinates. Thus U
R
∈ U (r) and V
i,R
∈ U (r
i
), and
Φ
−1
s
(R) = (π(R), (U
R
,V
1,R
,...,V
n,R
))
is smooth. The identity π◦ Φ
s
= pr
U
is immediate. Finally,
Φ
s
((o,g)· h) = s(o)· (gh) = (s(o)· g)· h = Φ
s
(o,g)· h,
so Φ
s
is G-equivariant.
Corollary 4.8 (Vertical tangent space). For R = (X,W )∈R
reg
,
kerdπ
R
={(Xξ,−η
1
W
1
,...,−η
n
W
n
) : ξ ∈ u(r), η
i
∈ u(r
i
)}.
In particular, relative to the framed gauge action, the vertical tangent space is canonically identified
with the Lie algebra
g = u(r)⊕
n
M
i=1
u(r
i
).
Proof.The local trivialization of Theorem 4.7 identifies the fiber throughRwith the rightG-orbit.
Differentiating the orbit map at the identity gives
δX = Xξ, δW
i
=−η
i
W
i
.
Becauseπis a submersion, its kernel is the tangent space to the fiber. Freeness makes the
infinitesimal-action map from g injective, so these directions exhaust kerdπ
R
.
Remark 4.9 (Scope of the bundle statement). Theorem 4.7 applies only to the framed minimal
realization space over one regular fixed-rank stratum. It does not assert a single bundle across
rank-changing boundaries, nonregular POVMs, varying outcome number, continuous outcome spaces,
or unused ancillary sectors. Those extensions may require stratified or groupoid-based geometry.
16
5 Complete-POVM Robustness Theory
5.1 Setting and norms
Let H
S
be a complex Hilbert space of dimension d, and let
E = (E
1
,...,E
n
)
be a finite POVM:
E
i
≥ 0,
n
X
i=1
E
i
= I.
For a density operator ρ and an event A⊆ I :={1,...,n}, define
E
A
:=
X
i∈A
E
i
and the operational typicality
T(ρ,E;A) := Tr(ρE
A
).
In the unitary-refinement program this trace formula represents the already-derived projective
typicality after purification and Naimark realization; it is not taken here as an independent
probabilistic axiom.
Definition 5.1 (Complete-measurement distances). For statesρ,σand two labeled finite POVMs
E = (E
i
) and F = (F
i
), define
d
st
(ρ,σ) :=∥ρ− σ∥
1
,
d
max
(E,F ) := max
i
∥E
i
− F
i
∥
∞
,
and
d
Σ
(E,F ) :=
n
X
i=1
∥E
i
− F
i
∥
∞
.
Define the event norm
d
event
(E,F ) := sup
A⊆I
∥E
A
− F
A
∥
∞
.
For a fixed partition Γ ={B
1
,...,B
m
}, define
d
Γ
(E,F ) := max
α
∥E
B
α
− F
B
α
∥
∞
.
The event norm gives the sharp measurement term for uniform control over the fixed labeled
event algebra. The sum and maximum distances provide convenient computable upper bounds:
d
event
(E,F )≤ d
Σ
(E,F )≤ nd
max
(E,F ).
The partition distance can be much smaller when fine-outcome perturbations cancel inside fixed
coarse blocks.
Remark 5.2 (Trace-norm convention). Throughout this section,∥·∥
1
is the unnormalized trace norm.
We do not include the conventional factor 1/2 used in the quantum-information trace distance.
17
5.2 Joint state-and-measurement robustness
Theorem 5.3 (Single-event joint perturbation bound). Letρ,σbe density operators and letE,F
be labeled finite POVMs. For every event A⊆ I,
|T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥
1
+∥E
A
− F
A
∥
∞
.
Consequently,
|T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥
1
+
X
i∈A
∥E
i
− F
i
∥
∞
.
Proof. Insert and subtract Tr(σE
A
):
Tr(ρE
A
)− Tr(σF
A
) = Tr((ρ− σ)E
A
) + Tr(σ(E
A
− F
A
)).
Because 0≤ E
A
≤ I, one has ∥E
A
∥
∞
≤ 1. Trace duality and ∥σ∥
1
= 1 give
|Tr((ρ− σ)E
A
)|≤∥ρ− σ∥
1
and
|Tr(σ(E
A
− F
A
))|≤∥E
A
− F
A
∥
∞
.
The second bound follows from the operator-norm triangle inequality.
Corollary 5.4 (Uniform event robustness). Under the hypotheses of Theorem 5.3,
sup
A⊆I
|T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥
1
- d
event
(E,F ).
Consequently,
sup
A⊆I
|T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥
1
- d
Σ
(E,F )
and
sup
A⊆I
|T(ρ,E;A)− T(σ,F ;A)|≤∥ρ− σ∥
1
- nd
max
(E,F ).
Proof. Apply Theorem 5.3 and take the supremum over A. The remaining bounds follow from
d
event
(E,F )≤ d
Σ
(E,F )≤ nd
max
(E,F ).
Remark 5.5. The factornis a worst-case consequence of controlling each effect separately. It should
not be interpreted as optimal. The event-level norm∥E
A
− F
A
∥
∞
or a partition-adapted norm is
preferable when available.
5.3 Fixed coarse graining
Let Γ ={B
1
,...,B
m
} be a fixed partition of the outcome labels. Define the coarse POVM
E
Γ
α
:= E
B
α
.
18
Proposition 5.6 (Coarse-event robustness). For every coarse event C ⊆{1,...,m},
T(ρ,E
Γ
;C)− T(σ,F
Γ
;C)
≤∥ρ− σ∥
1
+
X
α∈C
E
Γ
α
− F
Γ
α
∞
.
Hence
sup
C
T(ρ,E
Γ
;C)− T(σ,F
Γ
;C)
≤∥ρ− σ∥
1
- md
Γ
(E,F ).
Proof. Apply Theorem 5.3 to the coarse POVMs E
Γ
and F
Γ
.
Remark 5.7. Coarse graining is not merely a change of notation. A fine-measurement perturbation
can be large ind
Σ
while the induced coarse perturbation remains small because signed operator
changes cancel within a block.
5.4 Canonical normalization of approximately complete families
Let
F = (F
1
,...,F
n
), F
i
≥ 0,
be a labeled positive family with frame operator
S :=
n
X
i=1
F
i
.
Assume S is positive definite. Define the canonical normalized effects
N
S
(F
i
) := E
i
= S
−1/2
F
i
S
−1/2
.
Proposition 5.8 (Exact normalization and coarse-graining compatibility). The normalized family
E = (E
i
) is a POVM. For every fixed block B ⊆ I,
E
B
= S
−1/2
F
B
S
−1/2
, F
B
:=
X
i∈B
F
i
.
Thus canonical normalization commutes exactly with every fixed coarse-graining sum.
Proof. Positivity is preserved by congruence. Moreover,
X
i
E
i
= S
−1/2
X
i
F
i
!
S
−1/2
= S
−1/2
SS
−1/2
= I.
Linearity of finite summation gives the block identity.
Lemma 5.9 (Inverse-square-root deviation). If
∥S− I∥
∞
≤ δ < 1,
then
(1− δ)I≤ S ≤ (1 + δ)I,
S
−1/2
∞
≤ (1− δ)
−1/2
,
and
S
−1/2
− I
∞
≤ (1− δ)
−1/2
− 1.
19
Proof.The first inequality follows from the spectral characterization of the operator norm. Apply
the decreasing scalar functionx7→ x
−1/2
to the spectrum ofS. The maximal deviations occur at
1− δ.
Theorem 5.10 (Effect-wise normalization bound). Assume ∥S− I∥
∞
≤ δ < 1. Let
c
δ
:= (1− δ)
−1/2
− 1.
Then for every i,
∥E
i
− F
i
∥
∞
≤∥F
i
∥
∞
c
δ
(1− δ)
−1/2
- 1
.
Equivalently,
∥E
i
− F
i
∥
∞
≤∥F
i
∥
∞
δ
1− δ
.
The same estimate holds with F
i
,E
i
replaced by any coarse sums F
B
,E
B
.
Proof. Set A = S
−1/2
. Then
AF
i
A− F
i
= (A− I)F
i
A + F
i
(A− I).
Therefore
∥E
i
− F
i
∥
∞
≤∥A− I∥
∞
∥F
i
∥
∞
(∥A∥
∞
- 1).
Apply Lemma 5.9. Finally,
(1− δ)
−1/2
− 1
(1− δ)
−1/2
- 1
=
δ
1− δ
.
The coarse statement follows from Proposition 5.8.
Corollary 5.11 (Typicality cost of normalization). For every state ρ and event A⊆ I,
|Tr(ρE
A
)− Tr(ρF
A
)|≤∥F
A
∥
∞
δ
1− δ
.
Proof. Use trace duality, ∥ρ∥
1
= 1, and Theorem 5.10 for the coarse sum A.
5.5 Perturbing an entire normalized architecture
Consider two positive families F = (F
i
) and G = (G
i
) with frame operators
S =
X
i
F
i
, T =
X
i
G
i
,
and normalized POVMs
E
i
= S
−1/2
F
i
S
−1/2
, H
i
= T
−1/2
G
i
T
−1/2
.
Assumption 5.12 (Standing uniform frame-gap hypothesis for this section). Fix a constanta >0,
independent of the perturbation pair, such that
S ≥ aI, T ≥ aI.
20
Lemma 5.13 (Inverse-square-root Lipschitz bound). Under Assumption 5.12,
S
−1/2
− T
−1/2
∞
≤
1
2a
3/2
∥S− T∥
∞
.
Proof. Use the standard functional-calculus integral representation [14, 11]
X
−1/2
1
π
Z
∞
0
λ
−1/2
(X + λI)
−1
dλ
and the resolvent identity. Since S,T ≥ aI,
(S + λI)
−1
≤ (a + λ)
−1
and similarly for T. Hence
S
−1/2
− T
−1/2
∞
≤
∥S− T∥
∞
π
Z
∞
0
λ
−1/2
(a + λ)
2
dλ.
With the substitution λ = at,
Z
∞
0
λ
−1/2
(a + λ)
2
dλ = a
−3/2
Z
∞
0
t
−1/2
(1 + t)
2
dt.
The remaining integral is
B
1
2
,
3
2
Γ(1/2)Γ(3/2)
Γ(2)
π
2
,
so the original integral equals π/(2a
3/2
).
Theorem 5.14 (Complete normalized-POVM perturbation bound). Under Assumption 5.12, define
M
i
:= max{∥F
i
∥
∞
,∥G
i
∥
∞
}.
Then for every i,
∥E
i
− H
i
∥
∞
≤
1
a
∥F
i
− G
i
∥
∞
+
M
i
a
2
∥S− T∥
∞
.
Because
∥S− T∥
∞
≤
X
j
∥F
j
− G
j
∥
∞
,
on every subset satisfyingS,T ≥ aIandM
i
≤ M, theith normalized effect is Lipschitz in the labeled
positive family with an explicit constant depending on a and M.
Proof. Add and subtract S
−1/2
G
i
S
−1/2
:
E
i
− H
i
= S
−1/2
(F
i
− G
i
)S
−1/2
- (S
−1/2
− T
−1/2
)G
i
S
−1/2
- T
−1/2
G
i
(S
−1/2
− T
−1/2
).
Now
S
−1/2
∞
,
T
−1/2
∞
≤ a
−1/2
. Apply Lemma 5.13 to the final two terms. Their combined
coefficient is at most
2
1
2a
3/2
a
−1/2
= a
−2
.
Remark 5.15. The theorem controls simultaneous variation of every effect through the common
frame operators. This is stronger than applying an effect-wise perturbation estimate while ignoring
normalization coupling.
21
5.6 Smooth-path bounds
Let
t7−→ o(t) = (ρ(t),E(t))
be a C
1
operational path with a fixed outcome labeling and fixed coarse partition.
Proposition 5.16 (Pathwise typicality derivative). For every fixed event A,
d
dt
T(ρ(t),E(t);A) = Tr( ̇ρ(t)E
A
(t)) + Tr(ρ(t)
̇
E
A
(t)).
Consequently,
d
dt
T(ρ(t),E(t);A)
≤∥ ̇ρ(t)∥
1
+
̇
E
A
(t)
∞
.
Proof. Differentiate the trace product and use 0≤ E
A
(t)≤ I, trace duality, and ∥ρ(t)∥
1
= 1.
Corollary 5.17 (Integrated pathwise stability). For 0≤ s≤ t≤ 1,
|T(ρ(t),E(t);A)− T(ρ(s),E(s);A)|
≤
Z
t
s
∥ ̇ρ(τ )∥
1
+
̇
E
A
(τ )
∞
dτ.
Proof. Integrate Proposition 5.16 and apply the triangle inequality.
5.7 Scope and remaining quantitative problems
The trace, normalization, coarse-graining, and pathwise bounds proved above apply to arbitrary
finite-dimensional positive effects and therefore do not require rank-one measurements. Additional
hypotheses are needed only when one seeks quantitative control of support projectors, local support
frames, or realization-space lifts.
On a compact subset of a fixed-rank stratum, uniform lower bounds on the nonzero state and
effect spectra support local projector and frame estimates. Standard invariant-subspace perturbation
theory, including Davis–Kahan type control, provides the natural background for such spectral-gap-
dependent estimates [15]. Establishing explicit constants adapted to the present realization charts
remains separate. The next section nevertheless proves a compact-chart response bound of the form
∥ds
o
(u)∥
fr
≤ L
sec
∥ ̇ρ∥
1
+
X
i
̇
E
i
∞
!
,
with a finite chart-dependent constant; no explicit spectral-gap formula is asserted here.
Proposition 5.18 (Robustness alone does not determine a connection). The bounds in this section
constrain operational typicalities and the conditioning of canonical normalization. They do not, by
themselves, select a horizontal complement to kerdπ
R
in T
R
R
reg
.
Proof.
Every lift of a fixed operational tangent has the same first-order scalar typicality derivatives
because the typicality functions descend throughπ. The preceding bounds estimate those base-space
derivatives and finite differences, but they contain no criterion that distinguishes two lifts differing
by a vertical gauge vector. Selecting a connection therefore requires realization-level structure
beyond scalar robustness. The framed response geometry of Section 6 supplies that structure.
Remark 5.19. This section supplies operational and conditioning scales. Section 6 constructs the
transport law independently, and Section 7 establishes its informational canonicity within the
generated realization framework.
22
6 Robustness-to-Realization Response Geometry
The complete-POVM estimates of Section 5 control operational variation on the base manifold. This
section lifts that control to the framed realization bundle and establishes a baseline transport law.
The construction uses the simplest right-invariant Hilbert–Schmidt metric on the purification and
dilation frames. It is established independently of the informational canonicity argument developed
later.
6.1 Operational and framed response norms
For an operational tangent vector
u = ( ̇ρ,
̇
E
1
,...,
̇
E
n
)∈ T
o
O
reg
,
define
∥u∥
rb
:=∥ ̇ρ∥
1
+
n
X
i=1
̇
E
i
∞
.
The subscript “rb” identifies the robustness norm and distinguishes it from the smooth Hilbert–
Schmidt metric introduced earlier on the operational stratum.
For framed realization tangent vectors
v = (
̇
X,
̇
W
1
,...,
̇
W
n
), w = (
̇
Y ,
̇
Z
1
,...,
̇
Z
n
),
define
g
fr
R
(v,w) := Re Tr(
̇
X
†
̇
Y ) +
n
X
i=1
Re Tr(
̇
W
†
i
̇
Z
i
)
and
∥v∥
fr
:=
q
g
fr
R
(v,v).
Proposition 6.1 (Gauge invariance of the framed metric). For every g ∈ G and v,w ∈ T
R
R
reg
,
g
fr
R·g
d(R
g
)
R
v,d(R
g
)
R
w
= g
fr
R
(v,w).
Proof. For g = (U,V
1
,...,V
n
), the differential of the right action sends
(
̇
X,
̇
W
1
,...,
̇
W
n
)7−→ (
̇
XU,V
†
1
̇
W
1
,...,V
†
n
̇
W
n
).
Left and right multiplication by fixed unitaries preserve the Hilbert–Schmidt inner product. Summing
the purification and labeled dilation-block contributions gives the result.
6.2 Compact-chart realization response
Let s : U →R
reg
be any smooth local section supplied by Theorem 4.5.
Theorem 6.2 (Compact-chart realization-response bound). LetK⋐ Ube compact. Then there is
a finite constant L
K,s
0 such that
∥ds
o
(u)∥
fr
≤ L
K,s
∥u∥
rb
for every o∈ K and u∈ T
o
O
reg
.
23
Proof.The differentialdsis a continuous bundle map. In a finite-dimensional local trivialization,
the operator norm
o7−→∥ds
o
∥
rb→fr
is continuous. Compactness of K implies that its supremum is finite. Taking
L
K,s
:= sup
o∈K
∥ds
o
∥
rb→fr
gives the displayed estimate.
Corollary 6.3 (Pathwise section response). Leto: [a,b]→ KbeC
1
and setR
s
(t) :=s(o(t)). Then
̇
R
s
(t)
fr
≤ L
K,s
∥ ̇o(t)∥
rb
and
Length
fr
(R
s
)≤ L
K,s
Length
rb
(o).
Proof. The chain rule gives
̇
R
s
(t) = ds
o(t)
̇o(t). Apply Theorem 6.2 pointwise and integrate.
Remark 6.4 (Section dependence). The response of a chosen section is not gauge invariant. If
s
′
(o) =s(o)· g(o), thends
′
o
(u) contains an additional vertical term induced bydg
o
(u). Theorem 6.2
proves quantitative control for each smooth gauge choice; the next construction removes that
dependence.
6.3 Minimum-response connection
Recall the vertical tangent space
V
R
:= kerdπ
R
.
Define its framed orthogonal complement by
H
fr
R
:= V
⊥
g
fr
R
.
Theorem 6.5 (Gauge-invariant minimum-response connection). For everyR ∈ R
reg
andu ∈
T
π(R)
O
reg
, the affine lift space
L
R
(u) :={v ∈ T
R
R
reg
: dπ
R
(v) = u}
is nonempty and contains a unique vector of minimum framed norm. Denote it by
h
fr
R
(u) := arg min
v∈L
R
(u)
∥v∥
2
fr
.
Then:
- u7→ h
fr
R
(u) is linear;
- h
fr
R
(u)∈ H
fr
R
;
T
R
R
reg
= V
R
⊕ H
fr
R
;
4. R7→ H
fr
R
is smooth;
24
- for every g ∈ G,
d(R
g
)
R
H
fr
R
= H
fr
R·g
and
h
fr
R·g
(u) = d(R
g
)
R
h
fr
R
(u).
Consequently, H
fr
is a principal connection on
π :R
reg
→O
reg
.
Proof.
The submersion property of Corollary 4.6 makesdπ
R
surjective, soL
R
(u) is nonempty. The
positive-definite framed metric gives the orthogonal direct sum
T
R
R
reg
= V
R
⊕ V
⊥
g
fr
R
.
The restriction
dπ
R
H
fr
R
: H
fr
R
−→ T
π(R)
O
reg
is injective because its kernel isV
R
∩ H
fr
R
={0}. It is surjective because the orthogonal projection
of any lift ontoH
fr
R
has the same image underdπ
R
. Hence the restriction is a linear isomorphism,
and its inverse is the unique minimum-norm lift h
fr
R
.
The vertical tangent spaces form a smooth subbundle, and orthogonal complements with respect
to a smooth metric form a smooth subbundle. Proposition 6.1 shows that the right action preserves
the framed metric. Gauge invariance of π gives
d(R
g
)
R
V
R
= V
R·g
.
Therefore the right action carries framed orthogonal complements into framed orthogonal comple-
ments. Right equivariance of the minimum lift follows from uniqueness. These properties are exactly
the requirements for a principal connection.
Corollary 6.6 (Connection form and curvature). The horizontal distributionH
fr
determines a
principal connection one-form
ω
fr
∈ Ω
1
(R
reg
; g)
with
kerω
fr
= H
fr
.
Its curvature is
Ω
fr
= dω
fr
1
2
[ω
fr
,ω
fr
].
Proof.
This is the standard correspondence between smooth right-equivariant horizontal distributions
and principal connection one-forms.
6.4 Operational response geometry
Definition 6.7 (Minimum-response quadratic form). Foro ∈ O
reg
, choose anyR ∈ π
−1
(o) and
define
q
o
(u,w) := g
fr
R
h
fr
R
(u),h
fr
R
(w)
.
Write
q
o
(u) := q
o
(u,u).
25
Theorem 6.8 (Gauge-independent operational response geometry). The formq
o
is independent of
the representative R∈ π
−1
(o). It defines a smooth positive-definite quadratic form on TO
reg
.
Proof. Let R
′
= R· g. By Theorem 6.5,
h
fr
R
′
(u) = d(R
g
)
R
h
fr
R
(u).
Proposition 6.1 therefore gives
g
fr
R
′
h
fr
R
′
(u),h
fr
R
′
(w)
= g
fr
R·g
d(R
g
)h
fr
R
(u),d(R
g
)h
fr
R
(w)
= g
fr
R
h
fr
R
(u),h
fr
R
(w)
.
Thus the form descends to the base. Smoothness follows from smoothness of the horizontal inverse
of dπ. If u̸= 0, then h
fr
R
(u)̸= 0 because dπ
R
h
fr
R
(u) = u. Hence q
o
(u) > 0.
Theorem 6.9 (Robustness-to-response bound). Lets:U → R
reg
be a smooth local section and
K⋐ U compact. Then
q
q
o
(u)≤∥ds
o
(u)∥
fr
≤ L
K,s
∥u∥
rb
for every o∈ K and u∈ T
o
O
reg
. Consequently, there is a finite constant L
K
such that
q
q
o
(u)≤ L
K
∥u∥
rb
on K.
Proof. Differentiating π◦ s = id
U
gives
dπ
s(o)
ds
o
(u) = u.
Thusds
o
(u) is an admissible lift in the minimum definingh
fr
s(o)
(u). Minimum-norm optimality gives
q
q
o
(u) =
h
fr
s(o)
(u)
fr
≤∥ds
o
(u)∥
fr
.
Theorem 6.2 gives the second inequality. Set L
K
= L
K,s
.
Corollary 6.10 (Minimum-response path bound). For every C
1
operational path o : [a,b]→ K,
Z
b
a
q
q
o(t)
( ̇o(t))dt≤ L
K
Z
b
a
∥ ̇o(t)∥
rb
dt.
Proof. Apply Theorem 6.9 pointwise and integrate.
6.5 Status of canonicity
The connectionH
fr
is canonical relative to the framed Hilbert–Schmidt response geometry. It
is independent of local section, gauge equivariant, smooth, and introduces no branch weights
or regularization parameter. Theorem 6.5 therefore establishes relative canonicity before any
informational identity principle is introduced.
Relative canonicity alone does not identify the quantum-physical status of gauge-related al-
ternative lifts. Section 7 addresses that distinct question. It defines the generated informational
profile, proves pointwise and pathwise preservation under the established gauge action, states
Quantum Informational Identity, and derives absolute canonicity within the generated realization
framework. Scalar typicality derivatives are not used to select among lifts; they enter only as one
representation-invariant component of the generated profile.
26
6.6 Scope
The results in this section apply on one regular fixed-rank framed realization bundle. They do not
assert a single response connection across rank-changing boundaries, nonregular POVMs, changes
in outcome number, or other singular loci outside the regular framed model. Explicit spectral-gap
dependence of L
K,s
is also not derived here; only compact-chart finiteness is proved.
7Quantum Informational Identity and Profile-Relative Absolute
Canonicity
The preceding sections construct the framed response connection and prove its relative canonicity
independently of any informational identity premise. This section addresses the distinct interpretive
question of when gauge-related realization histories represent the same quantum-physical transport
content. To make that claim verifiable, the generated carrier, event algebra, and admitted covariance
group are now defined directly from the fixed framed architecture.
7.1 The fixed generated carrier and covariance structure
Definition 7.1 (Generated framed carrier and labeled event algebra). On a fixed regular stratum,
define the generated framed carrier
G :=K⊗A
and the fixed labeled projectors
Q
i
:= P
i
⊗ I
A
, i = 1,...,n.
For B ⊆ I, write Q
B
:=
P
i∈B
Q
i
. The labeled event algebra is
A := C
∗
(Q
1
,...,Q
n
)⊆B(G).
For every realization R∈R
reg
, the unified state |Ψ
R
⟩ defined in Section 3 lies in G.
Definition 7.2 (Admitted label-preserving covariance group). The admitted covariance group of
the generated framed architecture is
U
gen
:={Z ∈ U (G) : ZQ
i
Z
†
= Q
i
for every i}.
Equivalently,U
gen
is the unitary group of the commutant of the labeled event algebra. This group
is fixed on the entire regular stratum; it is not selected separately for each realization.
Proposition 7.3 (Gauge implementation preserves the generated architecture). Forg=
(U,V
1
,...,V
n
)∈ G, define
b
V
g
:=
n
M
i=1
V
†
i
!
⊗ U
T
∈ U (G).
Then, for every R∈R
reg
,
b
V
g
|Ψ
R
⟩ =|Ψ
R·g
⟩,
b
V
g
Q
i
b
V
†
g
= Q
i
,
b
V
g
A
b
V
†
g
= A,
and
b
V
g
U
gen
b
V
†
g
=U
gen
.
Proof.
The state identity is the implementation calculation from Proposition 3.9. The block-diagonal
factor
L
i
V
†
i
preserves each coordinate block ofK, whileU
T
acts only onA; hence
b
V
g
commutes
with everyQ
i
=P
i
⊗ I
A
. It therefore fixesAunder conjugation and belongs toU
gen
. SinceU
gen
is a
group, conjugation by
b
V
g
∈U
gen
maps it onto itself.
27
7.2 Generated informational profiles
Definition 7.4 (Generated realization system). For R∈R
reg
, define
G
R
:=
G,|Ψ
R
⟩,{Q
i
}
n
i=1
, A,U
gen
, T
R
.
Thus the carrier, labeled projectors, event algebra, and admitted covariance group are fixed by
the framed architecture; realization dependence enters through|Ψ
R
⟩and the represented event
assignment T
R
.
Definition 7.5 (Generated informational profile). The generated informational profile Prof
gen
(R)
consists exactly of:
- the state ω
R
(A) =⟨Ψ
R
|A|Ψ
R
⟩ on A;
- the labeled fine- and coarse-event structure i7→ Q
i
and B 7→ Q
B
;
- the finite event correlations ω
R
(A
1
···A
k
) for A
j
∈ A;
- the admitted covariance action (Z,A)7→ ZAZ
†
, together with Z|Ψ
R
⟩, for Z ∈U
gen
;
- the induced event values T
R
(B).
No claim is made about informational variables outside this finite-dimensional generated profile.
Definition 7.6 (Generated profile isomorphism). The profiles ofRandR
′
are generated-profile
isomorphic if there exists V ∈ U (G) satisfying
V |Ψ
R
⟩ =|Ψ
R
′
⟩, V Q
i
V
†
= Q
i
for every i, VU
gen
V
†
=U
gen
,
and T
R
(B) = T
R
′
(B) for every labeled event B.
Theorem 7.7 (Preservation of the generated informational profile). IfVimplements a generated
profile isomorphism from R to R
′
, then
Prof
gen
(R)
∼
=
Prof
gen
(R
′
).
Proof. For A∈ A,
ω
R
′
(V AV
†
) =⟨Ψ
R
|A|Ψ
R
⟩ = ω
R
(A).
The identitiesV Q
i
V
†
=Q
i
preserve all fine and coarse labeled events and makeAd
V
a unital
∗-automorphism ofA. Products satisfy (V A
1
V
†
)···(V A
k
V
†
) =V(A
1
···A
k
)V
†
, so all listed finite
correlations are preserved. Conjugacy ofU
gen
preserves the admitted covariance action, and equality
of T
R
and T
R
′
preserves the induced event values.
Corollary 7.8 (Pointwise gauge-profile invariance). For every R∈R
reg
and g ∈ G,
Prof
gen
(R)
∼
=
Prof
gen
(R· g).
Proof.
UseV=
b
V
g
. Proposition 7.3 supplies the state, event, algebra, and covariance identities.
Proposition 3.9 supplies T
R
(B) = T
R·g
(B).
28
Prof
gen
(R
′
) =⇒ R≡
phys
R
′
.
QII is a pointwise identity premise. It does not by itself identify entire transport histories.
Definition 7.10 (Dynamical generated informational profile). For a smooth realization path
e
γ : J →R
reg
, define
Prof
dyn
gen
(
e
γ) :=
t7→ Prof
gen
(
e
γ(t))
.
The fixed structuresG,{Q
i
},A, andU
gen
are part of every pointwise profile and need not be listed
a second time as a realization-dependent history.
Definition 7.11 (Isomorphism of dynamical generated profiles). Two dynamical generated profiles
are isomorphic if there exists a smooth mapV:J → U(G) such thatV(t) implements a generated
profile isomorphism at every t∈ J.
Theorem 7.12 (Pathwise preservation of the dynamical generated profile). Let
e
γ
1
,
e
γ
2
:J →R
reg
be smooth lifts of the same operational path and suppose
e
γ
2
(t) =
e
γ
1
(t)· g(t)
for a smooth gauge path g : J → G. Then
Prof
dyn
gen
(
e
γ
1
)
∼
Prof
dyn
gen
(
e
γ
2
).
Proof.
SetV(t) =
b
V
g(t)
. Smoothness ofgimplies smoothness ofV. Proposition 7.3 and Proposi-
tion 3.9 show that V (t) implements a generated profile isomorphism at every t.
Assumption 7.13 (Dynamical Quantum Informational Identity). Isomorphic dynamical generated
informational profiles represent the same quantum-physical transport content within the generated
realization framework.
Definition 7.14 (Dynamical Principle of Least Wasted Motion). The Dynamical Principle of Least
Wasted Motion states that two admissible realization histories with isomorphic dynamical generated
profiles cannot represent distinct quantum-physical transports within the generated framework.
Theorem 7.15 (Dynamical QII implies Dynamical LWM). Dynamical Quantum Informational
Identity implies the Dynamical Principle of Least Wasted Motion.
Proof.This is the direct specialization of Assumption 7.13 to admissible realization histories with
isomorphic dynamical generated profiles.
Lemma 7.16 (Smooth gauge relation between lifts). Let
e
γ
1
,
e
γ
2
:J →R
reg
be smooth lifts of the
same smooth operational path. Then there is a unique smooth map g : J → G such that
e
γ
2
(t) =
e
γ
1
(t)· g(t)
for every t∈ J. If the lifts share the same initial representative at t
0
, then g(t
0
) = e.
Proof.At eacht, the lifts lie in the same fiber, which is a free and transitiveG-orbit. This gives
a unique pointwiseg(t). In any local principal-bundle trivialization along the base path,g(t) is
obtained from the two smooth fiber-coordinate maps and is therefore smooth. Uniqueness makes
these local expressions agree on overlaps. The initial statement follows from freeness.
Corollary 7.17 (Pathwise quantum-physical identity). Under Dynamical QII, any two smooth
lifts of the same operational path represent the same quantum-physical transport content within the
generated framework.
29
7.4 Profile-relative absolute canonicity
Definition 7.18 (Profile-closed admissible transport). A realization transport is profile-closed
admissible if it is a smooth principal connection on the same framed realization bundle, preserves
the fixed outcome-label correspondence, and takes the dynamical generated profile to exhaust the
represented information-bearing content under comparison. A transport requiring an additional
represented variable is an extension of the theory, not a competing transport inside this profile-closed
class.
Definition 7.19 (Profile-relative absolute canonicity). A transport is profile-relatively absolutely
canonical if all profile-closed admissible principal connections determine the same quantum-physical
transport content, while a stated response geometry selects a unique representative of that common
content by a specified variational criterion.
Theorem 7.20 (Profile-relative physical uniqueness and minimum-response selection). Assume
Dynamical Quantum Informational Identity. On each regular fixed-rank framed realization bundle,
every profile-closed admissible principal connection determines the same quantum-physical transport
content for a fixed operational path and initial realization. Relative to the framed Hilbert–Schmidt
response geometry,H
fr
is the unique minimum-response representative of that content. HenceH
fr
is profile-relatively absolutely canonical in the sense of Definition 7.19.
Proof.
LetH
′
be a profile-closed admissible principal connection. Fix a smooth operational path and
a common initial realization, and let
e
γ
′
and
e
γ
fr
be the lifts determined byH
′
andH
fr
. Lemma 7.16
supplies a unique smooth gauge path relating them. Theorem 7.12 identifies their dynamical
generated profiles, and Dynamical QII identifies their represented transport content. SinceH
′
was
arbitrary within the profile-closed class, the content conclusion holds for every such connection.
By Theorem 6.5, every affine lift space contains a unique vector minimizing the framed Hilbert–
Schmidt norm, and these minimizers defineH
fr
. ThusH
fr
is the unique minimum-response
representative relative to the stated response geometry.
Remark 7.21 (Pointwise versus dynamical roles). QII identifies pointwise profile-isomorphic realiza-
tions. Dynamical QII, not pointwise QII alone, supports the transport-content conclusion. Neither
premise derives the framed Hilbert–Schmidt metric.
Remark 7.22 (Claim boundary). The theorem is conditional on Dynamical QII and on closure of the
represented content under the explicitly defined generated profile. It does not exclude extensions
with additional observables, apply across singular rank-changing loci, or assert that informational
identity uniquely forces the Hilbert–Schmidt response geometry.
Corollary 7.23 (Canonicity chain). On each regular fixed-rank framed realization bundle,
Dynamical QII plus profile closure =⇒ unique represented transport content,
while the framed Hilbert–Schmidt response geometry gives the unique minimum-response representa-
tive H
fr
.
8 Discussion
The principal-bundle theorem, complete-POVM robustness, minimum-response connection, and
informational identity theorem now form one continuous argument. Exact representation freedom is
encoded by vertical gauge directions. Operational perturbations are controlled by the robustness
30
norm. The framed Hilbert–Schmidt response converts operational motions into unique minimum-
response lifts, whose squared norms descend to a positive response geometry on the operational
base.
The canonicity result has two distinct components. The geometric component proves that the
framed response functional has a unique minimum-response lift. The informational component now
defines the carrier and admitted covariance structure explicitly and proves that exact gauge alterna-
tives preserve the represented profile pointwise and pathwise. Pointwise QII addresses realizations;
Dynamical QII addresses histories. Under the stated profile-closure restriction, Dynamical QII
identifies gauge-related histories as the same represented transport content. Relative to the framed
Hilbert–Schmidt geometry, minimum response supplies the unique representative of that content.
Dynamical LWM summarizes this identity consequence; it does not derive the response metric.
The meaning of “absolute” is profile-relative and explicitly scoped. The theorem applies within
the finite-dimensional generated realization framework on each regular fixed-rank bundle, under
Dynamical QII, and among profile-closed admissible connections. It does not exclude extensions
with additional represented variables or claim one smooth structure across singular rank changes,
nonregular architectures, or continuous outcomes.
9 Conclusion
Regular state–POVM strata admit framed purification–dilation realization bundles with compact
gauge fibers. The realization projection is a smooth principal bundle, and its vertical tangent spaces
are precisely the infinitesimal purification- and labeled outcome-frame transformations.
Complete-POVM robustness controls simultaneous perturbations of states and measurement
architectures, including fixed coarse events, canonical normalization, and smooth paths. The framed
Hilbert–Schmidt response geometry defines unique minimum-response lifts. Their horizontal spaces
form a smooth principal connection, and their squared norms descend to a gauge-independent
positive response geometry. On compact charts, minimum response is bounded by the operational
robustness norm.
The generated informational profile is built on the fixed carrierK⊗A, the fixed labeled event
algebra, and the explicitly defined label-preserving covariance group. Exact gauge transformations
preserve this profile, and smooth gauge paths preserve its dynamical history. Under Dynamical
QII and profile closure, profile-isomorphic histories determine the same represented transport
content. Relative to the framed Hilbert–Schmidt geometry,H
fr
is the unique minimum-response
representative. Together these conclusions constitute profile-relative absolute canonicity within the
generated realization framework.
Program Linkage
Paper 1 develops the refinement-derived conditional representation theorem for the Born measure [16].
Paper 2 extends the unitary-refinement program to generalized quantum measurements through
purification–dilation representation invariance and finite robustness [17]. The present Paper 3
supplies regular operational and realization geometries, complete finite-POVM robustness, the
framed minimum-response connection, the generated informational profile, and the QII-based
absolute-canonicity theorem. Future work concerns extension beyond regular finite-dimensional
generated realization bundles rather than selection of a different transport within the present
framework.
31
References
[1]
Armin Uhlmann. Parallel transport and quantum holonomy along density operators. Reports
on Mathematical Physics, 24(2):229–240, 1986.
[2]Armin Uhlmann. Transition probability (fidelity) and its relatives. arXiv preprint
arXiv:1106.0979, 2011. Revised 2016.
[3]
Xu-Yang Hou and Hao Guo. Uhlmann geometry of fixed-rank density matrices: A fiber-bundle
approach. arXiv preprint arXiv:2609.21875, 2026.
[4]Michel Journée, Francis Bach, P.-A. Absil, and Rodolphe Sepulchre. Low-rank optimization on
the cone of positive semidefinite matrices. SIAM Journal on Optimization, 20(5):2327–2351,
2010.
[5]P.-A. Absil, Robert Mahony, and Rodolphe Sepulchre. Optimization Algorithms on Matrix
Manifolds. Princeton University Press, 2008.
[6]Bart Vandereycken. Low-rank matrix completion by riemannian optimization. SIAM Journal
on Optimization, 23(2):1214–1236, 2013.
[7] John M. Lee. Introduction to Smooth Manifolds. Springer, 2 edition, 2013.
[8]Mark A. Naimark. Spectral functions of a symmetric operator. Izvestiya Akademii Nauk SSSR.
Seriya Matematicheskaya, 4:277–318, 1940.
[9]Roberto Beneduci. Notes on naimark’s dilation theorem. Journal of Physics: Conference Series,
1638:012006, 2020.
[10]Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometry of algorithms with
orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2):303–353,
1998.
[11] Tosio Kato. Perturbation Theory for Linear Operators. Springer, 1995.
[12]Shoshichi Kobayashi and Katsumi Nomizu. Foundations of Differential Geometry, Volume I.
Interscience Publishers, 1963.
[13]
Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins University
Press, 4 edition, 2013.
[14] Rajendra Bhatia. Matrix Analysis. Springer, 1997.
[15]
Chandler Davis and W. M. Kahan. The rotation of eigenvectors by a perturbation. iii. SIAM
Journal on Numerical Analysis, 7(1):1–46, 1970.
[16]Michael Tolbert. Typicality and unitary refinement: A conditional representation theorem for
the born measure. Version 3.1 manuscript, 2026.
[17]
Michael Tolbert. Gauge freedom and the born rule: Extending the unitary refinement program
to generalized quantum measurements. Version 3 manuscript, 2026.
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