Optimal Low-Rank Quantum State Tomography with Bounded-Sample Joint Measurements

1 Introduction

• Low-rank tomography improves on unrestricted Θ(d^2/ε^2) tomography because rank-at-most-r states require Θ(dr/ε^2) samples with unrestricted joint measurements.Pure states illustrate the extreme r = 1 case, requiring only order d samples.Source: §1 · p. 3

Related work

• Related work established Θ(dr^2/ε^2) for single-sample low-rank tomography and left the rank-dependent adaptive bounded-joint-measurement complexity open.The paper resolves this interpolation, including the adaptive t = 1 case and the unrestricted r = d case.Source: §1 · pp. 3–5

1 Introduction

• Bounded-sample joint measurements interpolate between separate measurements and fully joint strategies while retaining classical adaptivity across measurement rounds.Each round measures at most t fresh samples and retains only classical information.Source: §1 · pp. 3–4
• For sufficiently small ε, two theorems characterize the rank-dependent sample-complexity interpolation up to universal constants, with arbitrary classical adaptivity in the lower bound and a matching nonadaptive upper bound.The optimal unrestricted complexity is O(dr/ε2), attained when measuring order r2 samples jointly; this joint-measurement scale is necessary and sufficient.Source: §1 · pp. 4–5
• The lower bound controls Fisher-information trace for support variations, extends it through an adaptive score chain rule, and applies van Trees to obtain a trace-norm bound.The construction uses states with fixed uniform spectrum and varying support.Source: §1 · p. 5
• The upper bound averages unbiased matrix estimates from a Gaussian joint measurement, controls support and off-support errors, and projects onto rank-at-most-r density matrices.Gaussian structure supplies an operator-norm bound outside the state’s support.Source: §1 · p. 5

Organization

• The paper gives a proof overview, formalizes notation and the adaptive measurement model, proves lower and upper bounds, and concludes with further questions.These topics appear in Sections 2 through 6, respectively.Source: Organization · p. 7

2.1 Lower bound

• The lower bound applies van Trees to support-rotation parameters, converting bounded Fisher information into trace-norm loss; a depolarizing reduction handles ranks r>d/2.The parameter family has dimension Θ(dr), uses a smoothly truncated Gaussian prior, and operator-norm localization converts Frobenius loss to trace-norm loss.Source: §2.1 · p. 7, §2.1.1 · pp. 7–8, §2.1.4 · p. 12

2.1.2 Fisher information along an adaptive decision tree

• Adaptive protocols form decision trees whose transcript Fisher information obeys a chain rule, so nodewise bounds combine under the pathwise constraint Σ_i t_i(H_i)≤n.Each node selects a fixed POVM on at most t fresh samples, allowing fixed-measurement Fisher bounds to apply despite adaptivity.Source: §2.1.2 · p. 8

2.1.3 The hard family and the information at one node

• The hard family fixes the uniform spectrum and varies only the support via small rotations, yielding rank-r states with 2r(d−r) real parameters.The states are ρ_X=Π_X/r, and accurate tomography would provide an accurate estimator of the support parameter X.Source: §2.1.3 · pp. 8–9
• Every joint POVM on t samples has bounded Fisher-information trace, with the resulting nodewise estimate transferred from X=0 to arbitrary X by a unitary contraction.The proof treats single-sample and joint measurements separately, and for t>r^2 uses classical–quantum Fisher information and SLD additivity.Source: §2.1.3 · pp. 9, 11

2.1.4 Putting the bounds together

• Combining the transcript bound with van Trees yields expected trace-norm loss Ω(ar), contradicting a sufficiently accurate estimator and establishing the lower bound for r≤d/2.Confidence amplification and bounded-operator-norm postprocessing convert constant-probability tomography guarantees into controlled expected loss; the large-rank case follows by embedding and mixing.Source: §2.1.4 · p. 12

2.2 Upper bound

• The upper-bound protocol repeatedly applies a Gaussian joint measurement, averages unbiased Hermitian estimates, and projects onto density matrices of rank at most r; it is nonadaptive.It measures s=min{t,r^2} samples jointly per repetition, reflecting that larger joint blocks do not improve the asymptotic error bound.Source: §2.2 · p. 13, §2.2.1 · p. 13

2.2.2 Reducing the error to three blocks

• The projection analysis reduces estimation error to controlling ΠEΠ and Π⊥EΠ in Frobenius norm, plus Π⊥EΠ⊥ in operator norm.Here E=Y−ρ and Π is an analysis-only projector onto an r-dimensional subspace containing the support of ρ.Source: §2.2.2 · p. 14

2.2.3 Properties needed from one joint measurement

• The Gaussian measurement supplies an exact second-moment identity and expected column count O(√s), giving an s^-1/2 variance gain for the Frobenius-controlled blocks.For the complementary block, conditional on the measurement outcome, the columns outside the support remain independent standard complex Gaussian vectors, enabling separate operator-norm control.Source: §2.2.3 · p. 14

2.2.4 Why the joint measurement has these properties

• The measurement is a valid POVM because the nested support subspaces yield projectors whose differences sum to identity, while pseudoinverse factors integrate each layer to its projector.The construction uses the subspaces Vℓ,s and the reference Gaussian measure’s full support.Source: §2.2.4 · p. 15
• The antisymmetrizer tail decreases geometrically once q is a sufficiently large constant multiple of √s, yielding Eρ[J] = O(√s).This controls the distribution of the measurement’s layer index J.Source: §2.2.4 · p. 15
• The joint measurement has unbiased output, with Eρ[Y] = ρ, and its second moment follows from Gaussian permutation expansions and centering identities.The calculation relies on swap operators, auxiliary output factors, and cancellations of permutations fixing centered factors.Source: §2.2.4 · pp. 15–16

2.2.5 Putting the bounds together

• For fixed N, choosing s = min{t, r2} makes the sum of the first two terms reach order dr/N at s = r2; increasing s further changes it by at most a constant factor.Source: §2.2.5 · p. 16
• The same joint POVM is reused nonadaptively, achieving trace-norm error at most ε with probability at least 2/3 using the claimed sample count.Larger joint measurements do not improve the resulting asymptotic rate.Source: §2.2.5 · p. 17

3.1 States, matrix spaces, and measurements

• The preliminaries establish notation for quantum states, supports, projectors, positive semidefinite operators, and matrix spaces used in the tomography analysis.A quantum state is a positive semidefinite operator with unit trace, and the support of a positive semidefinite operator is its image, equivalently the orthogonal complement of its kernel.Source: §3.1 · p. 17

3.2 Tensor permutations and Gaussian matrices

• Tensor-permutation, partial-trace, pseudoinverse, and standard complex Gaussian-matrix notation is introduced for later constructions.The Gaussian measure γ_d,ℓ is defined from independent standard complex Gaussian entries.Source: §3.2 · pp. 18–19

3.3 Classical Fisher information

• Classical Fisher information is defined for differentiable parametric outcome laws, with Fisher information trace identified as the quantity used in the lower bound.The framework covers both continuous outcomes through densities and discrete outcomes through probability mass functions.Source: §3.3 · p. 19

3.4 Quantum Fisher information

• Quantum Fisher information is measurement-independent, upper-bounds the classical Fisher information of any parameter-independent POVM, and is additive under tensor-product samples.The section distinguishes the measurement-induced matrix I_M from the quantum matrix I_Q and introduces the corresponding joint-measurement notation for t samples.Source: §3.4 · pp. 20–21

3.5 Adaptive measurement protocols

• Adaptive protocols choose joint POVMs on at most t fresh samples using a private seed and previous classical outcomes, with bounded total sample complexity.Protocols may halt early, and their final estimate depends measurably on the seed and observed outcomes.Source: §3.5 · p. 21
• The theorem’s adaptive model permits only classical memory between rounds; persistent quantum memory is excluded because it could combine rounds into a larger coherent measurement.Error is measured in the Schatten 1 norm, up to universal constants relative to the conventional trace-distance normalization.Source: §3.5 · p. 22

Hard-family construction

• The proof begins with a local parameterization of a hard family of quantum states.Source: §4 · p. 22

Fisher information bounds

• Fisher information bounds are established for jointly measuring t samples, starting with the single-sample case.Source: §4 · p. 22

Adaptive lower bound

• The bounds are accumulated over adaptive transcript rounds and converted into expected trace-norm loss using the van Trees inequality.Source: §4 · p. 22

Large-rank regime

• A depolarizing-channel reduction completes the lower-bound argument in the large-rank regime.Source: §4 · p. 22

Hard family

• The hard states are rank-r maximally mixed states whose supports vary near a reference r-dimensional subspace while their eigenvalues remain fixed.The construction uses X with small operator norm, with X = 0 representing the reference support.Source: §4.1 · p. 22

Support parameterization

• Matrices X ∈ C^(d−r)×r tilt the reference support toward its orthogonal complement, and their columns define an orthonormal basis for the rotated support.The normalized construction produces an isometry, an orthogonal projector, and a rank-r density matrix.Source: §4.1 · p. 22

Local coordinates

• The real and imaginary parts of X supply 2r(d−r) real tangent coordinates at the reference state.This follows by differentiating the support family along X = sH at s = 0.Source: §4.1 · p. 23

Metric comparison

• For operator norms at most 1/4, parameter distance obeys ∥X − Y∥1 ≤ 2∥ΠX − ΠY∥1 = 2r∥ρX − ρY∥1.Thus nearby support parameters can be recovered from trace-norm estimates of the states.Source: §4.1 · pp. 23–24

4.2 Fisher information trace for joint measurements

• The proposition bounds the Fisher-information trace for every joint POVM on t samples and every parameter X, supplying the main technical input to the adaptive lower bound.The proof establishes the result at X = 0 and transfers it to all parameters.Source: §4.2 · p. 24

4.2.1 Single-sample measurements

• For single-sample measurements, block decompositions relative to S ⊕ S⊥ and positivity of the POVM density bound the Fisher-information trace through Cauchy–Schwarz.The likelihood derivatives are expressed using the off-diagonal block B_z; when Tr(A_z) = 0, positivity forces B_z = 0 and the contribution vanishes.Source: §4.2.1 · p. 25

4.2.2 Joint measurements on multiple samples

• For joint measurements with t ≤ r^2, cross terms are controlled by occupation-number operators, yielding the O(dt/r) Fisher-information trace bound.The construction produces a direction-independent positive semidefinite operator with small normalized trace, which bounds the one-excitation contributions.Source: §4.2.2 · pp. 26, 29, 30, 32
• For t > r^2, the general quantum Fisher-information bound gives the required O(dt) trace estimate.The argument uses symmetric logarithmic derivatives for the real and imaginary tangent directions.Source: §4.2.2 · pp. 26, 32
• For arbitrary X, a unitary U_X maps the support of ρ_X to the reference support, while a real-linear map L_X relates perturbation directions at X and 0.The construction uses the support isometry V_X and its orthogonal complement; projector differentiation shows the diagonal blocks vanish and yields the lower-left block defining L_X.Source: §4.2.2 · pp. 33–34
• Because L_X is a contraction, conjugating the measurement cannot increase the Fisher-information trace at X relative to the corresponding measurement at 0.The likelihood derivative identity transfers gradients through the adjoint of L_X, and the contraction property yields the trace comparison.Source: §4.2.2 · p. 35
• At X = 0, the proof combines the single-sample and multi-sample cases to establish the Fisher-information trace bound for joint POVMs.The multi-sample bound is obtained using the classical–quantum Fisher inequality and tensor-product additivity, with separate cases for t ≤ r^2 and t > r^2.Source: §4.2.2 · pp. 33, 35
• Applying the reference-point bounds to the conjugated measurement proves Equation (29) for every parameter X and every joint POVM on t samples.This completes Proposition 4.2 by reducing the arbitrary-X case to the already established X = 0 inequalities.Source: §4.2.2 · p. 35

Protocol transcript model

• Adaptive executions are represented as decision-tree paths, with each history selecting a POVM on at most t fresh samples and at most N nontrivial rounds.The protocol is padded with dummy rounds to a fixed length N; dummy rounds use no samples and contribute no Fisher information.Source: §4.3 · p. 35
• Parameter-independent reference measures and likelihood regularity ensure that conditional outcome densities factor into a valid transcript likelihood, including continuous outcomes.The random seed has a parameter-independent law and therefore contributes no Fisher-information term.Source: §4.3 · pp. 35–37

Adaptive Fisher chain rule

• The adaptive Fisher chain rule decomposes transcript Fisher information into the expected sum of roundwise conditional Fisher-information matrices.Cross terms vanish because each roundwise score has conditional mean zero given the preceding history, while diagonal terms follow from conditional Fisher information and total expectation.Source: §4.3 · p. 36

Adaptive Fisher information bound

• For the rank-r state family, Corollary 4.7 converts the chain rule and the pathwise sample bound into a Fisher-information bound for every adaptive protocol using at most t samples per round and N samples overall.Rounds measuring zero samples contribute zero information, while nonzero rounds are controlled by Proposition 4.2 before traces are taken.Source: §4.3 · p. 37

4.4 From a Fisher information bound to expected trace norm loss

• Van Trees converts limited transcript Fisher information into large expected squared estimation error, which becomes a trace-norm lower bound after restricting the estimator’s operator norm.The prior is supported on ∥X∥op < a, and the denominator includes both transcript Fisher information and the prior information I(π).Source: §4.4 · p. 37, §4.4.1 · p. 37

4.4.1 The van Trees inequality

• Van Trees applies to smooth compactly supported priors and regular likelihoods, with the adaptive transcript model satisfying the required regularity conditions.The inequality averages over θ drawn from π and the corresponding observation, and bounds estimation error using the likelihood Fisher information together with I(π).Source: §4.4.1 · pp. 37–38

4.4.2 A smooth probability density for the support parameter

• For p = 2mr rectangular support parameters, Lemma 4.9 constructs a smooth density supported within ∥X∥op < a with Fisher information at most C p d/a^2.The prior uses a Gaussian factor and smooth cutoff; the cutoff has gradient norm O(1/a), controlling its Fisher information contribution.Source: §4.4.2 · pp. 38, 40

4.4.3 From Frobenius loss to trace norm loss

• If both X and estimator T have operator norm at most a, their difference has operator norm at most 2a, converting the van Trees Frobenius bound into trace-norm loss.Lemma 4.10 applies this conversion to estimators satisfying ∥T∥op ≤ a almost surely under the constructed prior.Source: §4.4.3 · p. 40

4.5.1 From high-probability tomography to a support estimator

• Confidence amplification uses K = O(1 + log(1/δ)) independent repetitions and majority-radius postprocessing to obtain a 3η-accurate estimator with probability at least 1 −δ.The selected output is within 3η whenever more than half of the repetitions are successful, by Hoeffding’s inequality and the majority-radius argument.Source: §4.5.1 · p. 41
• Postprocessing the amplified tomographic output into Xa converts trace-norm accuracy into an expected-loss bound for estimating the support parameter under the required operator-norm constraint.The construction is measurable and applies to ranks 1 ≤ r ≤ d/2.Source: §4.5.1 · pp. 41–42

4.5.2 The lower bound when the rank is at most half the dimension

• For r ≤ d/2, van Trees combines the support-estimator loss bound with the Fisher-information bound to establish the adaptive lower bound on sample complexity for measurements acting on at most t samples per round.The argument applies for sufficiently small accuracy η and protocols using at most n samples on every execution.Source: §4.5.2 · p. 42

4.5.3 Rank lifting by depolarization

• Depolarization lifts rank-k states to rank-r states, halves pairwise trace distances, and simulates every adaptive lifted measurement protocol with identical transcript laws, round sizes, and sample counts.The adjoint channel transforms each lifted POVM into a valid POVM on the original states without changing the joint-measurement bound.Source: §4.5.3 · pp. 43–44
• The rank-lifting reduction applies the half-rank lower bound to the remaining regime and proves Equation (1) for all ranks.The proof sets k = ⌊r/2⌋ and concludes with c := c0/72.Source: §4.5.3 · pp. 44–45

5 Rank-sensitive tomography

• The protocol uses a Gaussian joint measurement M_t, designed to achieve the rank-sensitive tomography rate nonadaptively.The measurement is defined through Gaussian laws, support projectors, and pseudoinverse factors.Source: §5 · p. 45
• The operators of M_t form a valid POVM because their components integrate to nested support projectors that sum to the identity.The Born rule specifies the outcome density relative to the Gaussian reference measure.Source: §5 · p. 45

5.1 The protocol

• The complete protocol applies M_s independently B times to fresh sample blocks, averages the resulting Hermitian matrices, and projects onto rank-at-most-r density matrices.The final projection minimizes Frobenius distance over the compact set of rank-constrained states.Source: §5.1 · p. 46
• The measurement outcome is a pair (J, G), with 1 ≤ J ≤ min{d, t} and G ∈ C^{d×J}.The associated Hermitian matrix is used as the single-measurement estimator.Source: §5.1 · p. 46
• For every state, the associated Hermitian matrix is exactly unbiased and has an explicitly characterized uncentered tensor second moment.These identities provide the basis for analyzing repeated measurements and the resulting estimator.Source: §5.1 · p. 46
• For rank-at-most-r states, the expected outcome dimension satisfies Eρ[J] = O(min{r, t}), yielding a rank-dependent control on measurement fluctuations.The proposition also gives a corresponding bound for every Hermitian test matrix H.Source: §5.1 · p. 46
• Conditioned on J = ℓ, the components of G outside any support projector containing ρ are independent standard complex Gaussian vectors, independent of the supported component.This conditional Gaussian law is a key property used in the later correctness analysis.Source: §5.1 · p. 46

5.2.1 Support of the Gaussian moment operators

• The Gaussian moment operators have nested supports V_ℓ,t, with projectors commuting with tensor permutations and A^⊗t; their differences therefore define a valid POVM.The support projectors satisfy 0 = Π_0,t ⪯ Π_1,t ⪯ ··· ⪯ Π_ℓmax,t = I_d^⊗t, so the differences Δ_ℓ,t are orthogonal projectors summing to identity.Source: §5.2.1 · pp. 47–48

5.2.2 Bounding the mean of J

• If rank(ρ) ≤ r, every outcome with positive probability satisfies J ≤ r, because ρ^⊗t is supported inside V_r,t.The mean analysis uses antisymmetrizers to control the tail of J and establish this rank-dependent support cutoff.Source: §5.2.2 · pp. 50–51

5.2.3 The outcome law and its moments

• The Gaussian moment identity expresses Γ_ℓ,t through tensor permutations weighted by the cycle count c(π), implying that the Γ_ℓ,t commute pairwise and with every A^⊗t.The identity follows by applying complex Isserlis pairing to the Gaussian entries; surviving pairings correspond to permutations in S_t.Source: §5.2.3 · pp. 52–53
• Conditioned on J = ℓ, the estimator outcome is Y = (GG† − ℓId)/t, and Gaussian moment calculations determine its event contribution and conditional outcome law.The event density relative to γ_d,ℓ is Tr(Θ_ℓ,ρ(GG†)^⊗t), with total mass P_ρ[J = ℓ].Source: §5.2.3 · pp. 55–56
• The first- and second-moment identities follow by exploiting permutation invariance and separately evaluating ordered and coincident index pairs, including the t = 1 case.For t = 1, the ordered-pair contribution is empty; the resulting identity holds for all t ≥ 1.Source: §5.2.3 · pp. 56–57
• The measurement’s moment calculation yields a rank-dependent bound E_ρ[J] = O(min{r, t}) for states of rank at most r.The proof combines the moment identities with Lemma 5.5.Source: §5.2.3 · p. 57
• Conditional on J = ℓ, the component Π⊥G outside any support containing ρ is independent of ΠG and has the standard Gaussian law on Im(Π⊥).The conditional outcome density depends only on ΠG, so the orthogonal component retains its reference Gaussian distribution.Source: §5.2.3 · pp. 57–58

5.3 Analysis of the estimator

• The error analysis decomposes E = Y − ρ into support, off-diagonal, and orthogonal-complement blocks, using scalar variance bounds and a conditional Gaussian law.The decomposition is analytical only; the protocol does not need to know the support projector Π.Source: §5.3 · p. 58, §5.3.2 · p. 59

5.3.1 Projection onto low-rank states

• A rank-constrained Frobenius projection lemma combines orthogonal block decompositions and Cauchy–Schwarz to control the estimator’s projection onto low-rank states.The lemma applies when Π is a rank-r projector containing the support of ρ.Source: §5.3.1 · pp. 58–59

5.3.2 Bounds for the four blocks

• The four-block bounds use Equation (85) for supported and off-diagonal blocks and conditional independence of standard complex Gaussian matrices for the orthogonal-complement block.Conditioned on the measurement outcomes J1, ..., JB, the projected Gaussian matrices remain independent standard complex Gaussian matrices.Source: §5.3.2 · pp. 59, 61

5.3.3 Completion of the upper bound

• Choosing s = min{t, r^2} makes the first two error terms reach order dr/N at s = r^2, while larger joint measurements change the sum only by a constant factor.The analysis therefore explains why jointly measuring more than r^2 samples does not improve the resulting asymptotic rate.Source: §5.3.3 · p. 62

• Joint measurements improve single-sample tomography by at most a factor sqrt(t), reaching unrestricted collective complexity when t is order r^2.The matching upper and lower bounds establish this rank-dependent saturation, and the nonadaptive upper bound shows adaptivity improves complexity by at most constant factors.Source: §6 · p. 62
• The lower bound combines a universal Fisher-information trace bound for support rotations with a conditional score chain rule for adaptive protocols.This lets adaptive analyses reduce to bounds on each joint measurement, together with the van Trees inequality described in the paper context.Source: §6 · p. 62
• The upper-bound analysis controls supported and off-diagonal errors in Frobenius norm and off-support error in operator norm before applying rank-constrained projection.A second-moment identity handles the first bounds, the conditional Gaussian law handles the orthogonal-complement block, and the expected number of columns in G enters both norm estimates.Source: §6 · p. 62
• The Gaussian protocol’s relationship to earlier mixed-state tomography algorithms remains to be clarified.The discussion identifies this comparison as a direction for future work rather than resolving the precise relationship.Source: §6 · p. 62
• Constructing finite-outcome versions of the continuous-outcome protocol while preserving optimal sample complexity, and determining the required outcome count, remain open problems.Source: §6 · p. 62
• Persistent quantum memory invalidates roundwise fresh-sample size as the sole resource measure, motivating tradeoff analyses involving memory, sample complexity, and computational efficiency.The current results retain only classical information between rounds, whereas quantum memory can combine rounds into a larger coherent measurement.Source: §6 · pp. 62–63

Mathematical assistance

• GPT-5.5 and GPT-5.6 Sol substantially assisted proofs of Lemmas 4.3 and 4.4, which underpin the Fisher-information bound and t-dependent lower-bound sample complexity.The authors independently verified all mathematical arguments and references and take responsibility for the paper’s content.Source: §7 · p. 63
• The models helped reformulate earlier tomography algorithms and analyses without representation theory, while Section 5 provides the paper’s additional rank-dependent error analysis for the Gaussian formulation.The authors also used the models for general mathematical discussions, literature searches, and editorial revisions.Source: §7 · p. 63

A Statistical formalism

• The appendix fixes a dominated model with reference measures independent of the unknown state, enabling common likelihood derivatives and Fisher information calculations.It also records the density notation and regularity framework used in the main proof.Source: A Statistical formalism · p. 65

A.1 Dominated models and finite-dimensional POVMs

• Sigma-finiteness and absolute continuity ensure Radon–Nikodym likelihood densities exist, with null events having probability zero for every parameter value.These properties provide the fixed measure-theoretic foundation for subsequent differentiation.Source: §A.1 · p. 65
• A finite-dimensional POVM admits a measurable positive semidefinite operator density with respect to a state-independent probability measure dominating every state-induced outcome law.The density has trace d almost everywhere, and zero-likelihood sets may be assigned zero score and Fisher contributions.Source: §A.1 · p. 66

A.2 Adaptive transcript domination

• For adaptive protocols with pathwise sample bound N, jointly measurable conditional operator densities yield a parameter-independent measure that dominates every transcript law.Iterating the conditional Born rule constructs the global reference measure λ_N and the transcript likelihood for the support-rotation family.Source: §A.2 · pp. 66–67
• The adaptive densities are obtained by jointly measurable entrywise Radon–Nikodym derivatives, with martingale convergence supplying measurable limits and positivity enforced almost everywhere.Exceptional null sets are repaired without changing kernel integrals, preserving the required trace identity on every history slice.Source: §A.2 · pp. 66–67

A.3 Likelihood regularity

• If the state family is C1 in trace norm, conditional and full transcript likelihoods are C1 pointwise and in their respective L1 spaces, with jointly measurable derivatives integrating to zero.On compact parameter sets, the transcript likelihood satisfies 0 ≤ q_N,X ≤ d^t_N and |D_H q_N,X| ≤ N C_K d^t_N ||H||_F.Source: §A.3 · pp. 67–68

A.4 The van Trees inequality

• The appendix proves the van Trees inequality under the established likelihood regularity assumptions by combining score identities, integration by parts, compact prior support, and Cauchy–Schwarz.The proof handles zero denominators by setting corresponding quotients to zero and uses the vanishing boundary term from the compactly supported prior.Source: §A.4 · pp. 68–69

Measurable projection

• The upper-bound estimator therefore chooses a state nearest to its Hermitian matrix estimate in Frobenius norm, with ties resolved measurably.This makes the projection step well-defined as a measurable estimator.Source: B Measurable projection onto low-rank… · p. 70
• A Borel map selects, for every Hermitian matrix M, a rank-at-most-r state minimizing ||M − σ||F.The selection exists by the measurable maximum theorem, using compactness of the state set and continuity of the objective.Source: B Measurable projection onto low-rank… · p. 70