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Feb 16, 2023 456 likes •740 views

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

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Quantifying the incompatibility of quantum measurements relative to a basis Quantifying the incompatibility of quantum measurements relative to a basis

arXiv: 1901.06382 arXiv: 1901.06382

Georgios Styliaris Georgios Styliaris Paolo Zanardi Paolo Zanardi

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, where

“Basis”

States diagonal in

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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, where

Measurement

  • f observable #1

“Basis”

States diagonal in

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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SLIDE 4

, where

Measurement

  • f observable #1

“Basis”

Measurement

  • f observable #2

States diagonal in

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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SLIDE 5

, where

Measurement

  • f observable #1

“Basis”

Measurement

  • f observable #2

States diagonal in

“transition matrix” between two bases is bistochastic Output probability distributiton is more uniform than input one

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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SLIDE 6

Measurement

  • f observable #1

Measurement

  • f observable #2

States diagonal in

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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SLIDE 7

Measurement

  • f observable #1

Measurement

  • f observable #2

States diagonal in “Uniforming” classical post-processing

(bistochastic matrix)

Independent of

diagonal in

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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SLIDE 8

Notation: we write if and only if there exists a bistochastic matrix such that

“An orthogonal measurement

  • ver is more compatible

than over , relative to states diagonal in ”

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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Notation: we write if and only if there exists a bistochastic matrix such that

“An orthogonal measurement

  • ver is more compatible

than over , relative to states diagonal in ”

  • is a preorder over bases, i.e.,

i) (reflexivity) and ii) , imply (transitivity)

Properties:

Resource theories!

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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Notation: we write if and only if there exists a bistochastic matrix such that

“An orthogonal measurement

  • ver is more compatible

than over , relative to states diagonal in ”

  • is a preorder over bases, i.e.,

i) (reflexivity) and ii) , imply (transitivity)

Properties:

  • “Measurement over is more

compatible than over any other basis” Resource theories!

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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Notation: we write if and only if there exists a bistochastic matrix such that

“An orthogonal measurement

  • ver is more compatible

than over , relative to states diagonal in ”

  • is a preorder over bases, i.e.,

i) (reflexivity) and ii) , imply (transitivity)

Properties:

  • , where is any basis

mutually unbiased to

“Measurement over is more compatible than over any other basis” “Measurement over any basis is more compatible than over any mutually unbiased one” Resource theories!

Incompatibility relative to : Orthogonal Measurements Incompatibility relative to : Orthogonal Measurements

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The “Quantum” version of The “Quantum” version of

Notation: we write if and only if there exists a bistochastic matrix such that

“Dephasing” or “Measurement” map

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The “Quantum” version of The “Quantum” version of

Notation: we write if and only if there exists a bistochastic matrix such that

“Dephasing” or “Measurement” map

can also be understood as

  • rthogonal measurement emulation via
  • ther such measurements
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Quantifying the incompatibility relative to Quantifying the incompatibility relative to

Preorder Monotones (functions from “bases” to non-negative Real) Compatibility measure Incompatibility measure

Notation: we write if and only if there exists a bistochastic matrix such that

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Quantifying the incompatibility relative to Quantifying the incompatibility relative to

Preorder Monotones (functions from “bases” to non-negative Real) Compatibility measure Incompatibility measure

Notation: we write if and only if there exists a bistochastic matrix such that

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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)

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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 relative to The more incompatible two bases are, the more coherence states have.

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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)

(uniform averaging over states diagonal in ) is a measure of incompatibility!

+ convertibility results in Resource Theories of Coherence

Incompatibility relative to The more incompatible two bases are, the more coherence states have.

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Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations

by Maasen & Uffink + Coles et al. Entropic Uncertainty

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Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations

by Maasen & Uffink + Coles et al. Entropic Uncertainty Quantum Fluctuations

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SLIDE 21

Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations

by Maasen & Uffink + Coles et al. Entropic Uncertainty Quantum Fluctuations

Incompatibility relative to

is a measure of incompatibility! (w.r.t. first argument)

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Incompatibility and Uncertainty Relations Incompatibility and Uncertainty Relations

by Maasen & Uffink + Coles et al. Entropic Uncertainty Quantum Fluctuations is a measure of incompatibility! (w.r.t. first argument)

Incompatibility relative to

is a measure of incompatibility! (w.r.t. first argument)

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, where

Measurement

  • f observable #1

“Basis”

Measurement

  • f observable #2

States diagonal in

“transition matrix” between two bases is stochastic Output probability distributiton is more uniform than input one

Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements

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Notation: we write if and only if there exists a stochastic matrix such that

“A measurement can be emulated by any kind of probabilistic post-processing from an measurement”

Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements

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Notation: we write if and only if there exists a stochastic matrix such that

is a preorder over POVMs

“A measurement can be emulated by any kind of probabilistic post-processing from an measurement” Preorder Monotones (functions from POVMs to non-negative Real) Compatibility measure Incompatibility measure

Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements

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Notation: we write if and only if there exists a stochastic matrix such that

is a preorder over POVMs

“A measurement can be emulated by any kind of probabilistic post-processing from an measurement” Preorder Monotones (functions from POVMs to non-negative Real) Compatibility measure Incompatibility measure Results:

i) Complete family of monotones: like orthogonal measurements, but convex + homogeneous functions ii) Get rid of basis-dependence: Reduces to “ is a parent measurement of “

Incompatibility relative to : Generalized Measurements Incompatibility relative to : Generalized Measurements

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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

arXiv: 1901.06382 arXiv: 1901.06382

Thank you! Thank you!