The Optimal Uncertainty Relation

Employing the lattice theory on majorization, the universal quantum uncertainty relation for any number of observables and general measurement is investigated. It is found that 1) the least bounds of the universal uncertainty relations can only be properly defined in the lattice theory; 2) contrary to variance and entropy, the metric induced by the majorization lattice implies an intrinsic structure of the quantum uncertainty; and 3) the lattice theory correlates the optimization of uncertainty relation with the entanglement transformation under local quantum operation and classical communication. Interestingly, the optimality of the universal uncertainty relation found can be mimicked by the Lorenz curve, initially introduced in economics to measure the wealth concentration degree of a society.

[1]  M. O. Lorenz,et al.  Methods of Measuring the Concentration of Wealth , 1905, Publications of the American Statistical Association.

[2]  W. Heisenberg.  Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik , 1927 .

[3]  H. P. Robertson.  The Uncertainty Principle , 1929 .

[4]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[5]  A. Uhlmann,et al.  Stochasticity and Partial Order: Doubly Stochastic Maps and Unitary Mixing , 1982 .

[6]  D. Deutsch.  Uncertainty in Quantum Measurements , 1983 .

[7]  Maassen,et al.  Generalized entropic uncertainty relations. , 1988, Physical review letters.

[8]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[9]  M. Plenio,et al.  Entanglement-Assisted Local Manipulation of Pure Quantum States , 1999, quant-ph/9905071.

[10]  Ugo Vaccaro,et al.  Supermodularity and subadditivity properties of the entropy on the majorization lattice , 2002, IEEE Trans. Inf. Theory.

[11]  Thierry Paul,et al.  Quantum computation and quantum information , 2007, Mathematical Structures in Computer Science.

[12]  Sergio Verdú,et al.  On the Interplay Between Conditional Entropy and Error Probability , 2010, IEEE Transactions on Information Theory.

[13]  Lukasz Rudnicki,et al.  Majorization entropic uncertainty relations , 2013, ArXiv.

[14]  S. Friedland,et al.  Universal uncertainty relations. , 2013, Physical review letters.

[15]  F. Hiai,et al.  Majorization and Singular Values , 2014 .

[16]  K. Życzkowski,et al.  Strong majorization entropic uncertainty relations , 2014, 1402.0129.

[17]  Patrick J. Coles,et al.  Improved entropic uncertainty relations and information exclusion relations , 2013, 1307.4265.

[18]  C. Branciard,et al.  Tight State-Independent Uncertainty Relations for Qubits , 2015, 1512.02383.

[19]  C. Qiao,et al.  Equivalence theorem of uncertainty relations , 2015, 1512.02781.

[20]  Jun-Li Li,et al.  Reformulating the Quantum Uncertainty Relation , 2015, Scientific Reports.

[21]  R. Werner,et al.  Uncertainty relations for angular momentum , 2015, 1505.00049.

[22]  Yunlong Xiao,et al.  Improved uncertainty relation in the presence of quantum memory , 2016, 1604.04944.

[23]  Yunlong Xiao,et al.  Optimal Universal Uncertainty Relations , 2016, Scientific Reports.

[24]  Markus P. Mueller,et al.  A Generalization of Majorization that Characterizes Shannon Entropy , 2015, IEEE Transactions on Information Theory.

[25]  C. Macchiavello,et al.  Tight entropic uncertainty relations for systems with dimension three to five , 2017, 1701.04304.

[26]  G. Bellomo,et al.  Approximate transformations of bipartite pure-state entanglement from the majorization lattice , 2016, 1608.04818.

[27]  K. Życzkowski,et al.  Majorization uncertainty relations for mixed quantum states , 2017, 1709.10294.

[28]  R. Werner,et al.  State-Independent Uncertainty Relations and Entanglement Detection in Noisy Systems. , 2017, Physical review letters.

[29]  Patrick J. Coles,et al.  Entropic uncertainty relations and their applications , 2015, 1511.04857.

[30]  G. Bellomo,et al.  The lattice of trumping majorization for 4D probability vectors and 2D catalysts , 2018, Scientific Reports.

[31]  John Watrous,et al.  The Theory of Quantum Information , 2018 .

[32]  Chen Qian,et al.  State-independent uncertainty relations and entanglement detection , 2017, Quantum Inf. Process..

[33]  P. Alam.  ‘K’ , 2021, Composites Engineering.

[34]  P. Alam.  ‘A’ , 2021, Composites Engineering: An A–Z Guide.