The sandwich of two approximately commuting projective measurements #
Two projective measurements {X_a} and {Z_b} on one party that approximately commute on a
state cannot in general be measured jointly. The sandwich
R_{a,b} = Z_b X_a Z_b
always can: it is a POVM --- exactly, by projectivity of Z alone --- so its Naimark dilation is
a genuine projective joint measurement, and the dilation's compression identity carries every
Born-rule estimate about R to the dilated measurement unchanged.
What has to be proved is that the sandwich is approximately self-consistent across the two
parties, and that is the content of one_sub_sum_bornProb_sand_le: a five-link chain from
sum_{a,b} <psi| R_{a,b} (x) R'_{a,b} |psi> to 1,
whose links are, in order, the two parties' commutators (each paid for by one Cauchy--Schwarz
against a family of squared norms summing to at most one), the two parties' X-consistency
collapsed by sum_a X'_a = Id, and the Z-consistency. The one thing the chain must not do is
bound the q^2 outcome pairs one at a time: that would cost a factor q and the conclusion has
to be q-independent. Every link here is therefore an estimate on a single operator product ---
either a sum that telescopes exactly, or one Cauchy--Schwarz over the whole outcome set.
The statement carries an index iota with weights, because that is how it is used: the two
measurements come from a question and the bound is an average over questions, and averaging a
square root would otherwise need a second Jensen step.
The chain is proved once, for a bipartite model (BipartiteModel.one_sub_sum_bornProb_sand_le),
and the matrix chain is its instance in the tensor-product model. The dilation that turns the
sandwich into a projective joint measurement is a construction on the concrete registers here
(exists_projective_joint).
Weighted Cauchy--Schwarz #
Cauchy--Schwarz against a weight. The one inequality the chain's square roots come from,
and also the Jensen step that moves an average inside a square root (take v = 1).
Sums over a product of the outcome set #
In a model #
The chain is proved once, for a bipartite model with the two parties' projective measurements in
their algebras (BipartiteModel.one_sub_sum_bornProb_sand_le); the matrix statements further down
are its instances in the tensor-product model. The sandwich itself is defined in any ring
(sand), and is a POVM in any star-ordered one (POVMIn.sand).
A sum of mutually orthogonal idempotents is idempotent.
A sum of mutually orthogonal star projections is a star projection.
The sandwich R_{a,b} = Z_b X_a Z_b of two measurements.
Equations
- MIPRE.sand X Z p = Z p.2 * X p.1 * Z p.2
Instances For
The sandwich, as a POVM in a star-ordered ring.
Equations
- MIPRE.POVMIn.sand hX hZ = { mats := fun (p : A × A) => ⟨MIPRE.sand X Z p, ⋯⟩, nonneg := ⋯, normalized := ⋯ }
Instances For
A projection is a contraction on a unit vector.
Products of the two players' operators multiply factorwise.
The adjoint of a mixed product is the mixed product of the adjoints.
A product of commuting star projections of the two players is a star projection.
Cauchy--Schwarz for a mixed product, with the deviation on the first player's side.
Cauchy--Schwarz for a mixed product, with the deviation on the second player's side.
A contraction in front costs nothing.
The squared norms of a projective measurement's vectors sum to one.
The consistency of two projective measurements, as a defect from one.
The chain #
The ordered products' squared norms sum to exactly one.
Link 1: replacing each sandwich by the ordered product costs the first player's commutator.
Link 2: replacing the second player's sandwich by the ordered product costs the second player's commutator.
The diagonal agreement operator of two projective measurements is a star projection.
The disagreement operator of two projective measurements is a star projection.
The disagreement operator's expectation is the consistency defect.
Link 3: dropping the first player's X-outcome costs the X-consistency, and nothing that
depends on the number of outcomes: the whole outcome sum is one operator product.
Link 4: the second player's X-outcome sums away, exactly.
Link 5: the Z-agreement is one minus half the Z-consistency defect.
The chain, assembled #
The sandwich of two approximately commuting projective measurements is approximately
self-consistent across the two parties. The five links of the chain, added up. Every error term
is q-independent: the two commutators enter under a square root through one Cauchy--Schwarz
each, the X-consistency through a third, and the Z-consistency linearly.
Coarse-graining two projective measurements the same way costs nothing. For projective
families the summed deviation and the agreement determine each other exactly
(one_sub_sum_bornProb_eq), and agreement can only increase under a common coarse-graining
(sum_bornProb_le_map).
Quadratic forms: Cauchy--Schwarz and the order #
A projection is bounded by the identity.
The two factors #
Cauchy--Schwarz for a mixed product, with the deviation on Alice's side.
A contraction in front costs nothing, on Alice's side.
A contraction in front costs nothing, on Bob's side.
Sums of mutually orthogonal projections #
Products of the two factors #
Expanding a cross-party deviation.
The consistency of two projective measurements, as a defect from one. With both families projective the two diagonal sums are exactly one, so the agreement probability and the summed cross-party deviation determine each other.
The sandwich #
The sandwich is the Gram operator of the ordered product, which is where its positivity and its normalization both come from.
The sandwich is a POVM, exactly: only projectivity of Z and completeness of X are
used, and no approximate commutation whatsoever.
The sandwich, as a bundled POVM.
Equations
- MIPRE.sandPOVM hX hZ = { mats := fun (p : A × A) => ⟨MIPRE.sand X Z p, ⋯⟩, nonneg := ⋯, normalized := ⋯ }
Instances For
The chain #
Five links from the agreement of the two parties' sandwiches to one. The two middle links are the
parties' commutators, each paid for by one Cauchy--Schwarz against a family of squared norms
summing to at most one; the third collapses the X-outcome sum by completeness; the last is the
Z-consistency.
A projective measurement's elements are contractions.
The ordered products' squared norms sum to exactly one.
Link 1: replacing each sandwich by the ordered product costs Alice's commutator.
Link 2: replacing Bob's sandwich by his ordered product costs Bob's commutator.
Link 3: dropping Alice's X-outcome costs her X-consistency with Bob, and nothing that
depends on the number of outcomes: the whole outcome sum is one operator product.
Link 4: Bob's X-outcome sums away, exactly.
Link 5: the Z-agreement is one minus half the Z-consistency defect.
The chain, assembled #
The sandwich of two approximately commuting projective measurements is approximately
self-consistent across the two parties. The five links of the chain, added up. Every error term
is q-independent: the two commutators enter under a square root through one Cauchy--Schwarz
each, the X-consistency through a third, and the Z-consistency linearly.
The two-sided extension, and the dilated joint measurement #
Replacing Bob's operator inside a cross-party deviation.
The two-sided extension of a state: one ancilla register in the fixed state |a0>
adjoined to each party. This is def:expanded-state's shape with an inert ancilla, and it is
what a pair of Naimark dilations --- one per party --- is compressed back by.
Equations
- MIPRE.extVec2 ψ a₀ b₀ = (Matrix.kroneckerMap (fun (x1 x2 : ℂ) => x1 * x2) (MIPRE.ancillaEmbed dA a₀) (MIPRE.ancillaEmbed dB b₀)).mulVec ψ
Instances For
The Born rule on the twice-extended state is the Born rule of the two compressions.
An operator with an inert ancilla compresses to itself.
Bob's inert-ancilla operator has the same state norm on the extended state.
The dilated joint measurement #
Replacing Bob's operator inside a squared cross-party deviation, at the usual factor two.
The dilated pair's self-consistency is the sandwiches' agreement defect, exactly: the compression identity transports the whole Born-rule expansion.
The same bound against the undilated sandwich on Bob's side, extended by identity on his ancilla.
The theorem #
Two approximately commuting projective measurements on each party admit a joint projective measurement, on each party's space enlarged by one ancilla register, which is self-consistent across the parties and consistent with both ordered products of the originals --- all at errors independent of the number of outcomes.
The construction is the Naimark dilation of the sandwich Z_b X_a Z_b; hkA and hkB record
that the dilations compress back to it, which is what carries every estimate.
Coarse-graining #
Coarse-graining two projective measurements the same way costs nothing. The
state-dependent distance has no data-processing inequality, and a per-fibre triangle inequality
would cost the fibre size; what makes this free is that for projective families the summed
deviation and the agreement probability determine each other exactly
(one_sub_sum_bornProb_eq), and agreement can only increase under a coarse-graining.
A deviation as the norm of one vector #
sum_avg_norm_fibre_sq is a statement about vectors; this is the bridge to the cross-party
deviation, and the linearity that lets a fibre sum of vectors be read as the deviation of the
coarse-grained operators.
The fibre sum of the deviation vectors is the deviation of the coarse-grained operators.