The Mišta–Fiurášek qudit partial-estimation trade-off

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

$$ \sqrt{F-\frac1{N+d}}= \sqrt{(d-1)\left(\frac{N+1}{N+d}-G\right)}+ \sqrt{N\left(G-\frac N{N+d}\right)}. $$

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, $N\ge1$, $d\ge2$ | 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 ${n:\mathrm{Fin}(d)\to\mathbb N\mid\sum_i n_i=N}$ 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 $\binom{N+d-1}{d-1}$ | 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. | | $N/(N+d-1)\le G\le(N+1)/(N+d)$ | 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.

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 $d\ge3$. Completeness of that screen is ASSUMED-UNVERIFIED.

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.

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.

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 $1.23\times10^{-7}$. These finite numerical comparisons are separate from the theorem for all $N,d$; they provide no uniform numerical error theorem.

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 $N\ge1$, $d\ge2$ | 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 $1.23\times10^{-7}$. | | Eight citing works and Rapčan's thesis | Literature reading recorded in #16777; completeness ASSUMED-UNVERIFIED | None of the screened sources settles Eq. (2) for $d\ge3$. |

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.

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.