Quantifying the incompatibility of Quantifying the incompatibility of quantum measurements quantum measurements relative to a basis relative to a basis Georgios Styliaris Paolo Zanardi Georgios Styliaris Paolo Zanardi arXiv: 1901.06382 arXiv: 1901.06382
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements States diagonal in “Basis” , where
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements Measurement of observable #1 States diagonal in “Basis” , where
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements Measurement Measurement of observable #2 of observable #1 States diagonal in “Basis” , where
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements Measurement Measurement of observable #2 of observable #1 States diagonal in “Basis” , where Output probability “transition matrix” distributiton is more between two bases uniform than input one is bistochastic
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements Measurement Measurement of observable #2 of observable #1 States diagonal in
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements Measurement Measurement of observable #2 of observable #1 States diagonal in “Uniforming” classical post-processing (bistochastic Independent of matrix) diagonal in
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements “An orthogonal measurement Notation: we write if and only if over is more compatible than over , relative to states there exists a bistochastic matrix such diagonal in ” that
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements “An orthogonal measurement Notation: we write if and only if over is more compatible than over , relative to states there exists a bistochastic matrix such diagonal in ” that Properties: ● is a preorder over bases, i.e., Resource theories! i) (reflexivity) and ii) , imply (transitivity)
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements “An orthogonal measurement Notation: we write if and only if over is more compatible than over , relative to states there exists a bistochastic matrix such diagonal in ” that Properties: ● is a preorder over bases, i.e., Resource theories! “Measurement over is more compatible than over any other i) (reflexivity) and basis” ii) , imply (transitivity) ●
Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements “An orthogonal measurement Notation: we write if and only if over is more compatible than over , relative to states there exists a bistochastic matrix such diagonal in ” that Properties: ● is a preorder over bases, i.e., Resource theories! “Measurement over is more compatible than over any other i) (reflexivity) and basis” ii) , imply (transitivity) ● ● , where is any basis “Measurement over any basis mutually unbiased to is more compatible than over any mutually unbiased one”
The “Quantum” version of The “Quantum” version of Notation: we write if and only if “Dephasing” or “ Measurement” map there exists a bistochastic matrix such that
The “Quantum” version of The “Quantum” version of Notation: we write if and only if “Dephasing” or “ Measurement” map there exists a bistochastic matrix such that can also be understood as orthogonal measurement emulation via other such measurements
Quantifying the incompatibility relative to Quantifying the incompatibility relative to Preorder Notation: we write if and only if there exists a bistochastic matrix such Monotones (functions from “bases” to non-negative Real) that Compatibility measure Incompatibility measure
Quantifying the incompatibility relative to Quantifying the incompatibility relative to Preorder Notation: we write if and only if there exists a bistochastic matrix such Monotones (functions from “bases” to non-negative Real) that Compatibility measure Incompatibility measure
Incompatibility and Quantum Coherence Incompatibility and Quantum Coherence “Relative entropy of coherence” Interpretation as: i) Disturbance by a measurement (statistical intepretation of KL divergence) ii) Rate of distillable coherence (Coherence as a Resource Theory)
Incompatibility and Quantum Coherence Incompatibility and Quantum Coherence Incompatibility relative to “Relative entropy of coherence” Interpretation as: The more incompatible two bases are, the i) Disturbance by a measurement more coherence states have. (statistical intepretation of KL divergence) ii) Rate of distillable coherence (Coherence as a Resource Theory)
Incompatibility and Quantum Coherence Incompatibility and Quantum Coherence Incompatibility relative to “Relative entropy of coherence” Interpretation as: The more incompatible two bases are, the i) Disturbance by a measurement more coherence states have. (statistical intepretation of KL divergence) ii) Rate of distillable coherence (Coherence as a Resource Theory) is a measure of incompatibility! (uniform averaging over states diagonal in ) + convertibility results in Resource Theories of Coherence
Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations Entropic Uncertainty by Maasen & Uffink + Coles et al.
Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations Entropic Uncertainty by Maasen & Uffink + Coles et al. Quantum Fluctuations
Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations Entropic Uncertainty Incompatibility relative to by Maasen & Uffink + Coles et al. is a measure of incompatibility! (w.r.t. first argument) Quantum Fluctuations
Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations Entropic Uncertainty Incompatibility relative to by Maasen & Uffink + Coles et al. is a measure of incompatibility! (w.r.t. first argument) Quantum Fluctuations is a measure of incompatibility! (w.r.t. first argument)
Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements Measurement Measurement of observable #2 of observable #1 States diagonal in “Basis” , where “transition matrix” Output probability between two bases distributiton is more is stochastic uniform than input one
Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements “A measurement can be Notation: we write if and only if emulated by any kind of probabilistic post-processing there exists a stochastic matrix such from an measurement” that
Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements “A measurement can be Notation: we write if and only if emulated by any kind of probabilistic post-processing there exists a stochastic matrix such from an measurement” that Preorder is a preorder over POVMs Monotones (functions from POVMs to non-negative Real) Compatibility measure Incompatibility measure
Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements “A measurement can be Notation: we write if and only if emulated by any kind of probabilistic post-processing there exists a stochastic matrix such from an measurement” that Preorder is a preorder over POVMs Monotones Results: (functions from POVMs to non-negative Real) Compatibility measure i) Complete family of monotones: like orthogonal Incompatibility measure measurements, but convex + homogeneous functions ii) Get rid of basis-dependence: Reduces to “ is a parent measurement of “
Summary Summary ● We have introduced a notion of incompatibility for quantum measurements, relative to a reference basis ● Approach yields complete family of monotones, i.e., quantifiers of incompatibility ● Connection of incompatibility , quantum coherence and uncertainty relations ● Generalization to arbitrary POVM measurements Thank you! Thank you! arXiv: 1901.06382 arXiv: 1901.06382
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