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Majorization entropic uncertainty relations

Zbigniew Puchała, Łukasz Rudnicki, Karol Życzkowski

arXiv:1304.7755v3quant-phcs.ITmath.IT

TL;DR

Optimal entropic uncertainty bounds remain unresolved for dimensions N ≥ 3, while the Maassen–Uffink bound is not generally optimal. This paper uses majorization to derive explicit Rényi-entropy bounds, which outperform Maassen–Uffink for most sampled N = 5 unitaries.

  • Problem

    The optimal entropic uncertainty relation remains open for N ≥ 3, and the Maassen–Uffink bound is not generally optimal in higher dimensions.

    Source: §I · p. 1, §VI · p. 6

  • Method

    Majorization yields explicit lower bounds for sums of Rényi entropies, using maximal spectral norms of submatrices of the unitary matrix.

    Source: §II · p. 2

  • Results

    P = 0.984 for N = 5: the bound is stronger than Maassen–Uffink for that share of sampled Haar-random unitaries.

    Source: §IV · p. 4

  • Takeaways & Limitations

    The bounds are invariant under dephasing and permutation, and the paper derives a classical Maassen–Uffink analogue for stochastic transition matrices.

    Source: §VI · p. 6

Abstract

from arXiv · show

Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of R\'enyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. Obtained bounds are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. Numerical simulations show that for a generic unitary matrix of size N = 5 our bound is stronger than the well known result of Maassen and Uffink (MU) with a probability larger than 98%. We also show that the bounds investigated are invariant under the dephasing and permutation operations. Finally, we derive a classical analogue of the MU uncertainty relation, which is formulated for stochastic transition matrices.

I. INTRODUCTION

For two noncommuting observables, the entropies of their outcome distributions obey a lower bound depending on the unitary matrix connecting their eigenbases. This work uses majorization to derive explicit Rényi-entropy bounds for arbitrary order, aiming to improve on the Maassen–Uffink bound for generic unitaries.

I. INTRODUCTION

  • For a pure state measured in two observable eigenbases, the outcome distributions are linked by a unitary matrix, and their entropy sum has a state-independent lower bound depending on that matrix.The probabilities are squared overlaps with the respective eigenstates.Source: §I · p. 1
  • The optimal entropic uncertainty bound is known only for N = 2; for higher dimensions, optimality remains open and the Maassen–Uffink bound is not generally optimal.The open problem is stated for N ≥ 3.Source: §I · p. 1
  • Rényi entropy tends to Shannon entropy as α → 1, becomes min-entropy as α → ∞, and is non-increasing in α.The cited prior work reports an explicit α = 1/2 bound for N = 2.Source: §I · p. 2
  • The work derives lower bounds for the sum of two Rényi entropies of arbitrary order and reports that its bound is stronger than Maassen–Uffink for generic unitaries with high probability.The bounds are expressed using quantities dependent on the unitary matrix.Source: §I · p. 2
  • Majorization lets the approach use all entries of the unitary rotation matrix, while Rényi entropy’s Schur concavity converts majorization relations into entropy bounds.If x ≺ y, then Hα(x) ≥ Hα(y).Source: §I · p. 2

II. MAIN RESULT

The main result uses majorization of the joint probability vector and maximal-submatrix spectral norms to construct explicit bounds. Schur concavity then yields lower bounds for sums of entropies.

II. MAIN RESULT

  • Hα(p)+Hα(q) is bounded below by (1/(1−α)) ln[R1^α+(1−R1)^α] in the first simple bound.The bound follows from the majorization relation and the Schur concavity of Rényi entropy.Source: §II · p. 2
  • The coefficients are built from maximal m × n submatrices of U, selected by their largest spectral norm, with m+n=k+1.The spectral norm is the largest singular value; the coefficients satisfy s1≤s2≤…≤sN=1.Source: §II · p. 2
  • For s2, the maximum is over two-component row or column vectors; s3 also compares three-component vectors with 2 × 2 submatrices.The 2 × 2 comparison can depend on entry phases, not only moduli.Source: §II · p. 3, Figure 1 · p. 3
  • The corollaries extend the majorization result to any Schur-concave function and to Rényi entropies for every α≥0.Rényi entropy is Schur-concave, so majorization supplies the corresponding entropy inequality.Source: §II · p. 3
  • The joint probability vector p⊗q is majorized by Q=(R1, R2−R1, …, RN−RN−1), with the Rk defining its cumulative components.This is the majorization relation used to derive the bounds.Source: §II · pp. 2–4

III. EQUIVALENT UNITARY MATRICES AND ENTROPIC UNCERTAINTY

The section defines an equivalence between unitary matrices using permutations and diagonal unitaries. Under this equivalence, the entropic bounds are invariant, allowing analysis to be restricted to dephased matrices.

III. EQUIVALENT UNITARY MATRICES AND ENTROPIC UNCERTAINTY

  • Equivalent unitary matrices differ by row and column permutations and diagonal unitary operations.The equivalence is defined by the existence of permutation matrices P1, P2 and diagonal unitary matrices D1, D2.Source: §III · p. 4
  • Any unitary matrix is equivalent to one with real entries in its first row and first column; any 2 × 2 unitary is equivalent to a real rotation matrix.The former form is called dephased.Source: §III · p. 4
  • The probability distributions and bounds are invariant under the equivalence, reducing the analysis to dephased matrices of dimension N^2−2N−1.Permutations reorder coordinates, while diagonal unitaries leave the induced distributions unchanged.Source: §III · p. 4

IV. LOW DIMENSIONAL EXAMPLES

The section compares the proposed bounds with earlier uncertainty bounds for low-dimensional unitary families and examines their performance on random unitary matrices.

  • For N = 2, the examples use rotation matrices and present bound (20) for different Rényi parameters.Any 2 × 2 unitary matrix is similar to a rotation matrix, so this family suffices for the example.Source: §IV · p. 4
  • For N = 3, 4, and 5, the bounds are compared with Deutsch and Maassen–Uffink along families interpolating between the identity and circular-shift permutation matrices.The family is parameterized by β.Source: §IV · p. 4
  • For N = 3, the comparison examines whether the Maassen–Uffink bound is higher than B2 across the unistochastic subset of a bistochastic-matrix cross-section.The cross-section's light region includes its center, where the Fourier matrix F3 makes the Maassen–Uffink bound sharp.Source: §IV · p. 4
  • P = 0.984 for N = 5: the proposed bound is better than the Maassen–Uffink bound for this share of Haar-distributed random unitary matrices.The reported probabilities for N = 2, 3, 4, and 6 are 0.814, 0.971, 0.972, and 0.991, respectively.Source: §IV · p. 4

V. CLASSICAL ANALOGUES OF MU BOUNDS

The section formulates a classical analogue of the Maassen–Uffink relation for a probability vector transformed by a stochastic matrix.

  • The classical setup uses a stochastic matrix T to map a normalized probability vector P to another normalized vector P′.The stated conditions are Tij ≥ 0 and each column sums to 1.Source: §V · p. 4
  • The derivation invokes Słomczyński’s inequality for Shannon entropy and continues the estimate to obtain the classical bound.The quantity H(p)(T) is defined as a statistical mixture of columns of T weighted by the probabilities pi.Source: §V · p. 5
  • For any probability vector P and its image P′ = TP, the sum of their Shannon entropies is bounded below by the logarithm of the inverse of the largest entry of T.Source: §V · p. 5

VI. CONCLUDING REMARKS

The concluding remarks present majorization-based Rényi entropy bounds, report their comparison with the Maassen–Uffink bound, and identify applications and scope limits.

  • The optimal entropic uncertainty relation remains open for unitary rotations of size N ≥ 3.Source: §VI · p. 6
  • 97%: for random unitary matrices of size N = 3, the paper’s Shannon-entropy bound is stronger than the Maassen–Uffink bound in this share of cases.Source: §VI · p. 6
  • The majorization approach gives explicit bounds for Schur-concave functions, with components expressed using spectral norms, equivalently largest singular values, of maximal submatrices of U.The paper states that these bounds are directly computable.Source: §VI · p. 6
  • The bounds are invariant under unitary-matrix equivalences generated by permutation and dephasing operations.Source: §VI · p. 6
  • The bounds can be incorporated into the Gühne–Lewenstein bipartite entanglement criteria to provide sharpened criteria.Source: §VI · p. 6
  • The paper also gives a classical analogue for stochastic transition matrices: the Shannon-entropy sum for P and TP is bounded below by minus the logarithm of T’s largest entry.Source: §VI · p. 6
  • Friedland, Gheorghiu, and Gour independently developed related majorization-based uncertainty relations for Rényi entropies and Schur-concave functions [26].Their bounds can characterize generalized quantum measurements described by arbitrary numbers of POVMs.Source: §VI · p. 6

Appendix A: Useful lemma

Appendix A presents a lemma used as the main ingredient in the proof of Theorem 1, relating two orthonormal sets to the leading singular value of their overlap matrix.

Appendix A: Useful lemma

  • The lemma concerns two orthonormal vector sets and a rectangular matrix whose entries are their pairwise inner products, with its leading singular value used in the statement.The lemma also maximizes over normalized vectors.Source: Appendix A · pp. 6–7
  • When m = 1, the matrix norm reduces to the length of the corresponding vector.The passage identifies this as a special case of the matrix norm.Source: Appendix A · p. 7
  • A maximizing vector for the left-hand side of (A1) is constructed proportional to the sum |ξ0⟩ + |η0⟩.Source: Appendix A · p. 7
  • The notation lin{|1⟩, . . . , |m⟩} denotes the linear space spanned by those unit vectors.Source: Appendix A · p. 7
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