Metadata

slug: mista-fiurasek-2006-qudit-partial-estimation
bibkey: mista2006partialestimation
doi: 10.1103/PhysRevA.74.022316
url: https://arxiv.org/abs/quant-ph/0606112
triage: theorem
motivation_gids:
  - D5/S3/Quantum/Measurement/QuditPartialEstimationTradeoff.result

The Mišta–Fiurášek qudit partial-estimation trade-off

Problem

L. Mišta Jr. and J. Fiurášek, Optimal partial estimation of quantum states from several copies, arXiv:quant-ph/0606112v1, Phys. Rev. A 74, 022316, Introduction, Eq. (2), states:

Moreover, we also conjecture that the optimal fidelity trade-off for an ensemble of N identical pure states of a d-level system has the form

$F - \frac{1}{N + d} = (d - 1) (\frac{N + 1}{N + d} - G) + N (G - \frac{N}{N + d}) .$

Preregistration #16777 fixes the literal covariance-reduced PSD-seed model. The source is documented in the Library note.

Statement echo

Source clause	Lean counterpart	Fidelity
$N$ copies of a $d$ -level pure state, $N \geq 1$ , $d \geq 2$	`claim` quantifies natural `N`, `d` with precisely these two inequalities	Every allowed dimension and copy count is retained.
Symmetric input space tensored with one output qudit	The carrier is the product of the literal occupation subtype ${n : Fin (d) \to N ∣ \sum_{i} n_{i} = N}$ with `Fin d`	Occupation carriers occur inline; no alternate certificate space replaces them.
Eq. (12), the partial transpose of the symmetric projector	`OccupationPartialTransposeSpectral.RF`, `symmetricProjector`, `occupationEmbedding`	The normalized word-sector projector, partial transpose and compression are the definitions. Entry formulas are proved afterward.
Eq. (10), reference projection and output partial trace	`OccupationPartialTransposeSpectral.RG`, `outputReferenceProjector`, reused `PartialTraceMutualInformation.partialTraceRight`	The matrix product, partial trace and tensor with the output identity are retained.
Positive seed and Schur normalization	`chi.PosSemidef` and trace equal to the complex coercion of $(d - 1 N + d - 1)$	No rank restriction is imposed on competitors.
Output and estimation fidelities	Real parts of `trace (chi * RF N d)` and `trace (chi * RG N d)`	Both scores use the literal source operators.
$N / (N + d - 1) \leq G \leq (N + 1) / (N + d)$	The two bounds on real `G` in `claim`	The complete closed interval, including endpoints, is retained.
Optimal curve, Eq. (2)	`IsGreatest` of the set of all feasible output scores at fixed `G`	The displayed squared curve is both attained and an upper bound. Its square-root terms are nonnegative on the stated interval, so it is equivalent to Eq. (2).
Proof of the conjecture	`QuditPartialEstimationTradeoff.result : claim`	There are no additional hypotheses.

The result gives the exact information–disturbance frontier for all copy counts and qudit dimensions in the source’s PSD-seed optimization. The optimization includes every positive semidefinite seed with the specified trace and estimation score. An explicit rank-one seed attains each frontier point.

Gap

The source proves the qubit trade-off and conjectures Eq. (2) for general qudits. The literature screen recorded in #16777 covers eight citing works and Rapčan’s thesis, which describes the qudit case as a conjecture. No screened work settles Eq. (2) for $d \geq 3$ . Completeness of that screen is ASSUMED-UNVERIFIED.

Route

In OccupationPartialTransposeSpectral, the occupation indices are partitioned into coupled blocks and isolated singletons. BlockMatrices.weighted_block identifies each coupled restriction as an identity term plus a rank-one term and one distinguished coordinate. GlobalBound.global_operator_bound combines the exhaustive decomposition with the scalar Rayleigh estimate and monotonic domination by the all-reference block.

In QuditPartialEstimationTradeoff, private Scores.seed_support_bound transfers that matrix inequality to the literal seed fidelities. Private ScalarAssembly.scalar_frontier supplies the scalar supporting-line argument, including the endpoint. Private Attainment.frontierSeed_scores computes the attaining seed’s two fidelities; frontierSeed_psd and frontierSeed_trace give feasibility. result assembles the upper bound and attainment into IsGreatest.

Falsifier

A PSD seed with the specified trace and estimation score and an output score strictly above the curve would contradict result. A discrepancy between the source’s operators and the formal definitions would invalidate the model correspondence. The theorem does not independently establish the published covariance/Choi reduction from arbitrary instruments and Haar averages.

Evidence

The mathematical declarations are in OccupationPartialTransposeSpectral and QuditPartialEstimationTradeoff. The settling declaration is D5/S3/Quantum/Measurement/QuditPartialEstimationTradeoff.result; no atom or coverage claim is involved.

The four-slot escape audit is unfinished: #17100. It covers the new public theorems and the publicized or relocated owner declarations. Missing reconstruction, source-selection, bridge and sensitivity evidence is not a declared_validated registration.

Numerical check

The independent finite check is the experiment directory at commit cce08e4484a3406292ef841a8536a46d3ceafd0c. Command: OPENBLAS_NUM_THREADS=1 VECLIB_MAXIMUM_THREADS=1 python3 check.py from that directory; exit 0, ALL_OK. The script SHA-256 is 4ab2bfe6df891c3735e092664300c532c74ff26a75f3ef5941debdfc7f4de6b5.

For $d \in {2, 3, 4}$ and $N \in {1, 2, 3, 4}$ , the program constructs the source matrices and compares their maximal-eigenvalue support with the closed form and the curve support at 41 weights. The curve support uses a 4001-point grid. All 12 pairs pass; the largest reported deviation is $1.23 \times 1 0^{- 7}$ . These finite numerical comparisons are separate from the theorem for all $N, d$ ; they provide no uniform numerical error theorem.

Triage

What the settlement shows

Target item	Status and evidence kind	Result and boundary
Exhaustive occupation blocks, each rank one plus a coordinate term	Proved, kernel-checked: `OccupationPartialTransposeSpectral.BlockMatrices.weighted_block` and private `Global.weighted_decomposition`	Every coupled block and isolated singleton is included.
All-reference domination and scalar support reduction	Proved, kernel-checked: `OccupationPartialTransposeSpectral.GlobalBound.global_operator_bound`; private `QuditPartialEstimationTradeoff.Scores.seed_support_bound` and `ScalarAssembly.scalar_frontier`	The block bound is uniform in the occupation and the supporting parameter.
Attainment by a rank-one seed	Proved, kernel-checked: private `QuditPartialEstimationTradeoff.Attainment.frontierSeed_psd`, `frontierSeed_trace`, `frontierSeed_scores`, and public `result`	A feasible seed attains every allowed frontier score.
Eq. (2) for all $N \geq 1$ , $d \geq 2$	Proved, kernel-checked: `QuditPartialEstimationTradeoff.result`	This is the literal PSD-seed optimization, without an arbitrary-instrument formalization.
Single-copy and qubit cases	Proved as specializations of `result` at $N = 1$ and $d = 2$	The source identifies the $N = 1$ balance with Banaszek, PRL 86, 1366 (2001), reference [7], and proves the qubit curve as Eq. (1). Substitution into the displayed curve recovers those parameter cases; no separate operational theorem is claimed.
Published covariance/Choi reduction	Input from the source, Section II, Eqs. (3)–(14)	The reduction from arbitrary instruments to PSD seeds is not separately formalized.
Arbitrary-instrument operational statement	Open	Instruments, Haar averages and the covariance reduction require an additional formal layer.
Mixed-state and multi-output variants	Open	The theorem concerns identical pure inputs and one output qudit.
Finite matrix/support experiment	Computed: the pinned experiment entry above, command exit 0 and the stated SHA-256	Exactly the 12 pairs, 41 weights and 4001-point curve grid specified above; `ALL_OK`, maximum reported deviation $1.23 \times 1 0^{- 7}$ .
Eight citing works and Rapčan’s thesis	Literature reading recorded in #16777; completeness `ASSUMED-UNVERIFIED`	None of the screened sources settles Eq. (2) for $d \geq 3$ .

The argument supplies a general-dimension proof of the conjectured seed frontier. The source’s covariance reduction remains an input. No unformalized source result depending on that operational reduction is reclassified as kernel-checked here.

ASSUMED-UNVERIFIED

The literature screen is not exhaustive. Publication priority beyond the screened sources remains unverified. The instrument-to-seed reduction is a published premise of the model correspondence, not a theorem of these modules. The four-slot escape audit remains unfinished under #17100.

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