Weighted operators and variance families #
The operator (G_g)^{1/2} used throughout expansion.tex.
Uses CFC.sqrt (continuous functional calculus) to compute the matrix
square root of the PSD operator G.outcome g.
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- MIPStarRE.LDT.GlobalVariance.polynomialWeightSqrtOperator params G g = CFC.sqrt (G.outcome g)
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The weighted state |ψ_g⟩ = (I ⊗ √G_g)|ψ⟩, modeled as a density-matrix
transformation: ρ_g = W_g ρ W_g† where W_g = I ⊗ √(G_g).
This is not necessarily normalized — normalization would require dividing by
Tr(G_g ρ_B). We keep it unnormalized since the variance quantities in the
paper use unnormalized weighted expectations.
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The concrete operator A^u_{g(u)} for a fixed polynomial g.
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- MIPStarRE.LDT.GlobalVariance.pointConditionedOutcomeOperatorAtPolynomial params strategy g u = (strategy.pointMeasurement u).outcome (g.toFun u)
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The two-outcome point event selecting the answer g(u).
This is the submeasurement form of the point operator used in the first and last
steps of lem:local-variance-of-points (expansion.tex, lines 305 and 311).
Its some () outcome is exactly A^u_{g(u)}; the none outcome is the
complementary point-answer mass.
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The selected outcome of pointConditionedEventSubMeasAtPolynomial is
A^u_{g(u)}.
The paper's weighted operator A^u_{g(u)} ⊗ (G_g)^{1/2}
on the bipartite space d * d.
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The right-register intermediate I ⊗ (G_g)^{1/2} A^u_{g(u)}
from the first and last self-consistency moves of
lem:local-variance-of-points (expansion.tex, lines 306 and 310).
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The local variance of A(g) on the weighted state |ψ_g⟩.
Operators are lifted to the left tensor factor of the bipartite state.
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The global variance of A(g) on the weighted state |ψ_g⟩.
Operators are lifted to the left tensor factor of the bipartite state.
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The polynomial-averaged local variance of the conditioned points family.
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The polynomial-averaged global variance of the conditioned points family.
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The Option Unit event submeasurement selecting axis-line answers that
match g(u) at the queried point u on the left side of lem:generalize-b.
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The Option Unit event submeasurement selecting axis-line answers that
match the restriction of g to ℓ on the right side of lem:generalize-b.
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The residual projective Option Unit event from the proof of
lem:generalize-b: axis-line answers that collide with g at the sampled point
u, but are not the restricted polynomial g|_ℓ. After the
projective-measurement expansion, the squared difference is controlled by this
collision event.
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The residual event above, forgetting projectivity to a submeasurement. Keeping
this definition as the toSubMeas of generalizeBCollisionEventProjMeasAtPolynomial
prevents the projective and submeasurement views from drifting apart.
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The event operator for the residual line-collision event in lem:generalize-b.
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The event operator B^ℓ_{[f(u)=g(u)]}: sum of axis-line measurement
outcomes f that evaluate to the same value as g at point u.
Equations
- MIPStarRE.LDT.GlobalVariance.generalizeBLeftOperatorAtPolynomial params strategy g qu = (MIPStarRE.LDT.GlobalVariance.generalizeBLeftEventSubMeasAtPolynomial params strategy g qu).outcome (some ())
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The event operator B^ℓ_{[f = g|_ℓ]}: sum of axis-line measurement
outcomes f that agree with g restricted to line ℓ.
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The weighted left operator in lem:generalize-b
on the bipartite space d * d.
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The weighted right operator in lem:generalize-b
on the bipartite space d * d.
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