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The Complementary Information Principle of Quantum Mechanics

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Pith's one-line read The information gained from the first of two incompatible measurements fixes the tightest possible majorization bounds on the outcome distribution of the second, and those bounds can be computed by semidefinite programming.

desk verdict A genuinely new conditional majorization approach to uncertainty regions, but the one-step flatness formula for the upper bound does not obviously produce a valid probability vector and needs to be fixed. read the letter →

arxiv 1908.07694 v1 pith:BL3XJ7LE submitted 2019-08-21 quant-ph math-phmath.MP

Yunlong Xiao , Kun Fang , Gilad Gour This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81P4581P15 PACS 03.67.-a
keywords complementaryinformationprinciplemajorizationuncertaintyrelationssemidefiniteprogrammingquantummeasurementsregionsjointcausality
topics The Measurement Problem
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to show that complementarity between two incompatible measurements can be stated directly on the outcome probability vectors, without selecting an entropy or other scalar uncertainty measure. The claim is that once a first measurement returns a fixed outcome distribution p, every quantum state compatible with p produces a second-measurement outcome q that lies between two unique probability vectors r and t in the majorization order, and these are the tightest possible bounds. The paper proves that r and t can be computed efficiently with semidefinite programs, and uses them to build explicit outer approximations for the full uncertainty region of any reasonable uncertainty measure. If correct, this supplies a measure-independent benchmark for uncertainty relations and fundamental limits for all joint uncertainty measures.

What carries the argument

Majorization is the ordering that carries the argument: for probability vectors x and y, x≺y when the sum of the k largest entries of x does not exceed that of y for every k, equivalently the Lorenz curve of x lies below that of y. Comparison is done via marginal majorization, which fixes the pre-test vector p and orders post-test outcomes by majorization. The set of n-outcome probability vectors under this order is a complete lattice—every subset has a unique infimum and supremum—which is why r and t exist. The semidefinite programs compute, for each k, the smallest and largest possible value of the sum of the k largest post-test probabilities over all states satisfying the pre-test Born-rule constraints; these produce the lower envelope r and a raw upper envelope s. Because s need not be concave, the flatness process replaces it by the least concave curve above it, giving the optimal upper bound t.

What would settle it

Concretely, prepare a known state ρ, compute p from the Born rule, solve the two SDP families to obtain r and t, then measure N on many fresh copies and estimate the empirical post-test vector q; if the empirical Lorenz curve ever falls outside the envelope r≺q≺t beyond sampling error, the claimed bounds are wrong for that scenario.

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Extended reading notes

Core claim

The central discovery is Theorem 2: for a fixed pre-test measurement M and outcome distribution p, the set Q of post-test probability vectors has a unique greatest lower bound r and unique least upper bound t under majorization, so every q in Q satisfies r≺q≺t. Geometric language makes this vivid: the Lorenz curve of every admissible post-test distribution lies between the Lorenz curves of r and t, and the two curves form the tightest envelope allowed by the pre-test data. The lower bound r comes from minimizing, over all states compatible with p, the maximal sum of the k largest post-test probabilities for every k, while the upper bound t is obtained by taking the corresponding maxima and then applying a flatness procedure that restores the concavity a Lorenz curve must have. The paper presents this result as a complementary information principle: the classical information gained in the pre-test confines the uncertainty of the post-test in the tightest possible way, generalizing Heisenberg's complementarity.

Load-bearing premise

The argument assumes the black box produces independent, identically distributed copies of one quantum state and that the pre-test and post-test are performed on fresh copies, so the first measurement never disturbs the system that is later measured.

Editorial extensions

If this is right

  • For any pair of rank-one projective measurements and any pre-test outcome p, the post-test distribution of every compatible state lies in the majorization interval [r,t], and this interval is computable by parallel semidefinite programs.
  • The union of these intervals over all p yields an explicit outer approximation of the full uncertainty region for every non-negative Schur-concave uncertainty measure, and for qubits the approximation coincides with the true region.
  • Every joint uncertainty measure that is non-negative and monotone under doubly stochastic relabeling has state-independent lower and upper bounds given by the paper's formulas, tight for qubits.
  • The bounds extend to multiple post-test measurements and to POVMs, and give sufficient conditions for resource-theoretic state transformations, such as entanglement and coherence, when only partial information about the state is available.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next step, not taken in the paper, is to use the gap between r and t as a quantitative measure of remaining complementarity and to study how it shrinks as the pre-test measurement becomes more informative; this could sharpen information-causality statements beyond the two extreme cases the paper discusses.
  • Because the bounds are computed by semidefinite programs, they can serve as a common benchmark for the many published uncertainty relations: any candidate relation is at least as loose as the projection of [r,t], so the projection gives a fair way to compare different entropy-based bounds on one scale.
  • The flatness process shows the least upper bound is generally not attained by a physical state; characterizing the gap between t and the achievable set Q, and identifying the states that come closest to it, is an open convex-geometric question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper introduces a 'complementary information principle' for sequential black-box testing of two incompatible measurements. For a fixed pre-test measurement M with outcome distribution p, the set S(M,p) of states compatible with p defines a set Q of post-test probability vectors for measurement N. The authors claim that Q has a unique greatest lower bound r and least upper bound t under majorization, that both can be computed by semidefinite programs (Theorem 1), and that they are optimal because the majorization lattice is complete (Theorem 2). They then use r and t to outer-approximate uncertainty regions for arbitrary Schur-concave uncertainty measures, prove exactness for qubits, and derive state-independent bounds on general joint uncertainty measures, with comparisons to Maassen-Uffink and direct-sum/direct-product majorization bounds.

Significance. If the main theorem is correct, the paper provides a measure-independent characterization of conditional post-test uncertainty and a general method for outer-approximating uncertainty regions, with exact qubit results. The explicit SDP formulations, the analytic qubit bounds, and the detailed comparisons with Maassen-Uffink and direct-sum/direct-product majorization relations are useful and go beyond previous work that fixes a particular uncertainty measure. The conceptual move of conditioning on partial pre-test information instead of assuming full state knowledge is also worthwhile. However, the constructive specification of the optimal upper bound t is not fully justified as written, and this is load-bearing for Theorem 2 and all subsequent applications.

major comments (1)
  1. [Main text, Eq. (7); Supplemental Material, proof of Theorem 2] The one-step flatness construction in Eq. (7) does not, in general, produce a non-increasing probability vector, so the proof of Theorem 2 is incomplete for the t defined there. For example, applied to S=(0.4,0.1,0.2,0.3,0), the stated rule gives j=3, i=2, a=0.15, and hence T=(0.4,0.15,0.15,0.3,0), which is not in P_n. The proof of Theorem 2 uses t in P_n as a non-increasing vector and as the least concave majorant of the partial sums; it also asserts t=⋁Q only after arguing that t∈P_n. No argument is given that the particular s_k values arising from the quantum SDPs in Eq. (6) avoid this phenomenon for n≥4. The fix is local: the flatness step should be iterated until the vector is non-increasing, or the authors should prove that one pass is terminal for the special class of s_k from Eq. (6). Until then, the optimality of t, and therefore the universal uncertainty region and joint-uncertainty bounds that rely on t, are not established.
minor comments (5)
  1. [Main text, paragraph after Theorem 2] The claim that the bounds 'can be efficiently computed' and that s_k 'can still be solved efficiently via parallel computations even for large dimension n' is an overstatement: computing all s_k requires solving sum_{k=1}^n binom(n,k)=2^n-1 independent SDPs, and parallelization does not reduce the total computational work. The complexity statement should be restated honestly; this does not affect the mathematical validity of the bounds.
  2. [Supplemental Material, Remark 2] The printed s vector in the n=4 example sums to 1.0001 rather than 1; the rounding should be cleaned up or the numerical precision stated.
  3. [Main text, Eq. (12)] The simplification ~R_p(f,g)={(f(p),y)| g(t)≤y≤g(r)} is presented as an equality, but it implicitly assumes that every intermediate uncertainty value between g(t) and g(r) is achieved by some q with r≺q≺t. As written this is an outer approximation; a short justification or an explicit inclusion sign would be clearer.
  4. [Main text, opening of the framework] The phrase 'two independent and identically distributed resources ρ' is potentially confusing; it should be 'two independent copies of the same state ρ', since the derivation assumes the pre-test and post-test are performed on fresh copies rather than sequentially on one physical system.
  5. [Supplemental Material, proof of Theorem 2] In the contradiction argument, the proof writes inequalities for '∀l∈{1,...,i−1}' and '∀l∈{j,...,n}' even though l was fixed; this quantifier sloppiness should be corrected for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the SDP bounds are exact optimizations over the stated compatible-state set, and the claimed optimality rests on external lattice results rather than on fitted inputs or author-unique premises.

full rationale

Score 0. Theorem 1 computes each r_k as a minimization and each s_k as a maximization over the explicitly defined set S(M,p), with p fixed by Born's rule; no parameter is fitted to the post-test set Q and no predicted quantity is defined as the fitted value. The optimality claim in Theorem 2 is obtained from the external completeness of the majorization lattice (Rapat [103]; Bondar [104]; Bosyk et al. [105]) and the external flatness construction of Cicalese and Vaccaro [106], so it is not imported from a self-citation chain. The self-citations that appear ([13], [63], [88], [110], [113]) are used for background, benchmarks, definitions, a resource-theory review, or an in-preparation experimental claim; none carries the proof of Theorems 1-2, so under the stated rules they are normal citations and do not constitute circularity. Per the review rule, I explicitly flag two non-circular missing-support items. First, the main text (Eq. (7)) and the Supplemental Material 'Proof of Theorem 2' assert that a single flatness pass 'suffices' to construct the optimal upper bound t, but no proof is given that one pass yields a non-increasing probability vector or that no new violation appears; the construction can leave a vector outside P_n (e.g., increments (0.4,0.1,0.2,0.3,0) flatten once to (0.4,0.15,0.15,0.3,0)). Second, the statement 'The experimental data and our theoretical results fit well [113]' cites a paper 'in preparation' by overlapping authors, so that confirmation is not yet independently verifiable. Both are correctness or completeness concerns, not reductions of the claimed results to their inputs, so they do not affect the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted; the SDPs solve exact optimizations. The main imported background is the completeness of the majorization lattice, the Hardy-Littlewood-Pólya theorem, and the flatness algorithm. The 'complementary information principle' is a name for the framework, not a new physical entity. No invented mediators or forces appear.

assumptions (4)
  • standard math The probability simplex with majorization forms a complete lattice (Rapat 1991; Cicalese and Vaccaro 2002).
    Used in Theorem 2 to guarantee the existence of a unique infimum and supremum of Q under majorization and to justify the flatness process.
  • standard math Hardy-Littlewood-Pólya theorem: x≺y if and only if x = Dy for some doubly stochastic matrix D.
    Invoked when converting majorization bounds into bounds on any Schur-concave joint uncertainty measure J.
  • domain assumption The pre-test and post-test measurements are rank-one projective measurements on an n-dimensional Hilbert space, and the state set S(M,p) contains all density matrices with the fixed pre-test marginals p.
    Theorem 1's SDP formulations are written for projective measurements. The paper asserts extension to POVMs without a proof, so this is an unverified domain assumption.
  • standard math Born rule: p_j = <u_j|ρ|u_j> for the pre-test measurement outcomes.
    Foundation of the probability vectors p and q in the black-box testing scenario.

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Pith. "Pith review of The Complementary Information Principle of Quantum Mechanics." pith.science (2026). https://pith.science/paper/BL3XJ7LE

@misc{pith2026190807694,
  author       = {Pith},
  title        = {Pith review of: The Complementary Information Principle of Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BL3XJ7LE}},
  note         = {Machine review of arXiv:1908.07694}
}
read the original abstract

The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.

Figures

Figures reproduced from arXiv: 1908.07694 by the authors.

Figure 1
Figure 1. FIG. 1: (color online) Scenarios of preparational uncertainty rela [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: (color online) An example of Theorem [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: (color online) A schematic diagram depicts the uncer [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (4 more)
Figure 1
Figure 1. Figure 1: FIG. 1: (color online) A schematic diagram of lattice structure [PITH_FULL_IMAGE:figures/full_fig_p012_1.png]
Figure 2
Figure 2. Figure 2: FIG. 2: (colour online) Uncertainty regions with di [PITH_FULL_IMAGE:figures/full_fig_p015_2.png]
Figure 3
Figure 3. Figure 3: FIG. 3: (color online) Comparison of several known bounds via Lorenz curves. The example is taken as the probability [PITH_FULL_IMAGE:figures/full_fig_p016_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4: (color online) A schematic diagram depicts the classical range of [0 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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