PaperlayerView exact revision on arXiv

Source-linked AI summary

Universal Uncertainty Relations

Shmuel Friedland, Vlad Gheorghiu, Gilad Gour

arXiv:1304.6351v2quant-ph

TL;DR

The paper addresses why uncertainty should be quantified only by entropy, showing that relabelling invariance leads to Schur-concave uncertainty measures. It derives fine-grained majorization relations that yield uncertainty bounds for two bases and general POVMs, with an illustrative Shannon-entropy comparison to an existing bound.

  • Problem

    Recent uncertainty-relation work commonly uses Shannon and Rényi entropies, although the paper argues there is no fundamental reason to treat them as the most adequate quantifiers.

    Source: p. 1

  • Method

    The paper uses invariance under relabelling and monotonicity under random relabelling to identify non-negative Schur-concave functions as uncertainty measures, then derives majorization relations.

    Source: pp. 2–3

  • Results

    The fine-grained relations cover two orthonormal bases and general L POVMs; for c > 0.83, the Shannon bound overperforms around 90% of the time.

    Source: pp. 3–4

  • Takeaways & Limitations

    Applying arbitrary uncertainty quantifiers to the vector relation generates an infinite family of scalar uncertainty relations.

    Source: p. 4

    Extending the results to uncertainty relations with quantum memory is non-trivial and is left for future work.

    Source: p. 4

Abstract

from arXiv · show

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs \emph{entropic measures} to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, \textcolor{black}{significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A \textbf{84}, 052117 (2011)].} Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.

Supplementary Information Universal Uncertainty Relations

The supplementary information proves the results for two orthonormal bases and for L > 2 POVMs, and discusses weighted uncertainty relations.

Sec. A

Sec. B

Sec. C

Appendix A: Two orthonormal bases

Appendix A proves the uncertainty relation for two orthonormal bases by bounding partial sums and establishing results used to compute its components.

Proof of the two-basis relation

  • The proof bounds each sum of the k largest components of p(ρ) ⊗ q(ρ) by Ωk, establishing the majorization relation.Source: Appendix A · p. 6

Properties of Ωk

  • For orthonormal bases, choosing a basis-element state makes Ωk equal 1 for every d ≤ k ≤ d^2.Source: Appendix A · p. 6

Theorem 1

  • The appendix proves the maximization result by considering pure states and cases based on the overlap of the projectors' subspaces.The proof treats orthogonal subspaces separately from the case where the operator norm of their projector sum exceeds 1.Source: Appendix A · pp. 7–8

1. Exact expressions for Ω1 and Ω2

Theorem 1 and symmetry between the two measurement distributions are used to derive the expressions for Ω1 and Ω2.

Exact expressions for Ω1 and Ω2

  • Theorem 1 supplies an ingredient for computing Ω1 and Ω2.Source: §1 · p. 8
  • The derivation also uses symmetry under interchanging p and q.Source: §1 · p. 8
  • Combining the preceding relations yields the expression for Ω2 in equation (9) of the main text.Source: §1 · p. 8

2. The Upper Bounds eΩk

The appendix derives explicit upper bounds eΩk and uses them to prove a majorization uncertainty relation.

2. The Upper Bounds eΩk

  • Theorem 1 is used to prove exact expressions for the upper bounds eΩk defined in the main text.Source: §2 · p. 8
  • In the Young-like diagram, boxes represent components pmqn, and eΩk maximizes rectangle sums across D possible rectangles.The rectangle dimensions range from 1 × D to D × 1; the component probabilities are arranged lexicographically.Source: §2 · pp. 8–9, Figure 4 · p. 9
  • Because Ωk ≤ eΩk, the proof applies the partial-sum argument from the two-basis relation to establish the upper-bound relation.Source: §2 · p. 9

Appendix B: L ⩾2 POVMs

The paper extends its two-basis results to L ≥2 POVMs and uses upper bounds on Ω_k to construct a vector that satisfies the uncertainty relation. A three-basis example shows this relation is nontrivial where summing pairwise relations gives a trivial bound.

Appendix B: L ⩾2 POVMs

  • For L ≥2 POVMs, the paper extends the two-basis relation and constructs a vector satisfying relation (13) from the upper bound (16) on Ω_k.The paper does not provide an explicit expression for ω.Source: Appendix B · p. 9
  • The bounds eΩ_k are generally strict for POVMs because the technical theorem used for two measurements does not generalize to more than two.Unlike the POVM case, the first two upper bounds are equalities for two orthonormal bases.Source: Appendix B · p. 9
  • For three bases in dimension four, eΩ_1 ≅ 0.78 and eΩ_k = 1 for k > 1 yield a nontrivial relation, while pairwise relations give a trivial lower bound of zero.Each pair of bases shares a common eigenvector, but the three common eigenvectors are different.Source: Appendix B · pp. 9–10

Appendix C: Weighted uncertainty relations

The appendix incorporates unequal measurement weights into the uncertainty-relation formalism. It shows that the main vector relation yields weighted entropic uncertainty relations when measurements are repeated according to their weights.

Appendix C: Weighted uncertainty relations

  • The weighted relation generalizes the original entropic uncertainty relation to L measurements that may have different weights.The weights tℓ are probabilities associated with the measurements.Source: Appendix C · p. 10
  • The main vector relation yields weighted entropic uncertainty relations when measurements are not all distinct, with weights represented by measurement multiplicities.The appendix treats this weighted vector relation as a special case of the main result, not a new uncertainty relation.Source: Appendix C · p. 10
  • Applying an additive uncertainty measure to the vector relation produces a weighted scalar uncertainty relation, with tℓ = wℓ/W forming a probability distribution.The entropic relations considered use Shannon or Rényi entropy.Source: Appendix C · p. 10
Loading 1304.6351v2…