State independent Uncertainty Relations From Eigenvalue minimization P. Giorda, L. Maccone A. Riccardi QUit - GROUP, UNIVERSITY OF PAVIA Torino, May 2019
Outline Motivation 1 Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations First Result: Mapping onto eigenvalue problem 2 Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors Wrap up 3
Outline Motivation 1 Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations First Result: Mapping onto eigenvalue problem 2 Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors Wrap up 3
Uncertainty Relations Superposition principle ! Complementarity I Complementary Properties cannot have joint definite values for a given state | ψ i of the system I Preparation uncertainty E ff ects of Complementarity can be quantified depending on the aim, protocol, functional used e.g I Variance based uncertainity relations I Entropy based uncertainty relations In each framework the e ff ect of complementarity is described in a di ff erent way
Uncertainty Relations Superposition principle ! Complementarity I Complementary Properties cannot have joint definite values for a given state | ψ i of the system I Preparation uncertainty E ff ects of Complementarity can be quantified depending on the aim, protocol, functional used e.g I Variance based uncertainity relations I Entropy based uncertainty relations In each framework the e ff ect of complementarity is described in a di ff erent way
Uncertainty Relations Superposition principle ! Complementarity I Complementary Properties cannot have joint definite values for a given state | ψ i of the system I Preparation uncertainty E ff ects of Complementarity can be quantified depending on the aim, protocol, functional used e.g I Variance based uncertainity relations I Entropy based uncertainty relations In each framework the e ff ect of complementarity is described in a di ff erent way
Uncertainty Relations Hystorically URs have been thouroghly studied and di ff erent ways to quantify complementarity have been proposed: given two Observables A , B Heisenberg-Robertson | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � (Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929) Bialynicki-Birula (position and momentum), Deutsch (bounded operators), H ( A )+ H ( B ) � c ( A , B ) (Bia ł ynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)
Uncertainty Relations Hystorically URs have been thouroghly studied and di ff erent ways to quantify complementarity have been proposed: given two Observables A , B Heisenberg-Robertson | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � (Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929) Bialynicki-Birula (position and momentum), Deutsch (bounded operators), H ( A )+ H ( B ) � c ( A , B ) (Bia ł ynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)
Uncertainty Relations Hystorically URs have been thouroghly studied and di ff erent ways to quantify complementarity have been proposed: given two Observables A , B Heisenberg-Robertson | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � (Heisenberg, Z. Phys. 1927, H. Robertson Phys. Rev 1929) Bialynicki-Birula (position and momentum), Deutsch (bounded operators), H ( A )+ H ( B ) � c ( A , B ) (Bia ł ynicki-Birula, Mycielski 1975, Comm. Math. Phys; Deutsch 1983, Phys Rev Lett)
Uncertainty Relations The two approaches have di ff erent purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A , B | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing Entropy based URs � ! information about the SHAPE of probability distributions { p ( a n ) } , { p ( b n ) } H ( A )+ H ( B ) � c ( A , B ) independent on the “relabeling” of the outcomes I good candidates for protocols in which the amont of information about measurments outcomes is THE relevant issue I fundamental in protocols such as F quantum cryptography (PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)
Uncertainty Relations The two approaches have di ff erent purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A , B | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing Entropy based URs � ! information about the SHAPE of probability distributions { p ( a n ) } , { p ( b n ) } H ( A )+ H ( B ) � c ( A , B ) independent on the “relabeling” of the outcomes I good candidates for protocols in which the amont of information about measurments outcomes is THE relevant issue I fundamental in protocols such as F quantum cryptography (PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)
Uncertainty Relations The two approaches have di ff erent purposes, applications, possible caveats Variance based URs ! information about SPECTRUM OF A , B | h ψ | [ A , B ] | ψ i | 2 / 4 ∆ 2 A ∆ 2 B � I they can give trivial results I strongly dependent on the “relabeling” of the outcomes I this is also their usefulness in experiments and for theoretical purposes: F entenglement detection F spin squeezing Entropy based URs � ! information about the SHAPE of probability distributions { p ( a n ) } , { p ( b n ) } H ( A )+ H ( B ) � c ( A , B ) independent on the “relabeling” of the outcomes I good candidates for protocols in which the amont of information about measurments outcomes is THE relevant issue I fundamental in protocols such as F quantum cryptography (PJ Coles, M Berta, M Tomamichel, S Wehner - Reviews of Modern Physics, 2017)
Outline Motivation 1 Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations First Result: Mapping onto eigenvalue problem 2 Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors Wrap up 3
Sum uncertainty relations L. Maccone, A. K. Pati, PRL 113 (26), 260401 (2014) Given N A n bounded operators acting on H M N ∆ 2 ∑ V Tot ( | ψ i ) = | ψ i A n � l B n = 1 the problem of trivial results is removed the goal is now to find a STATE INDEPENDENT LOWER BOUND l B direct numerical minimization HARD for large dimension M Di ff ernt strategies have been proposed Analyitc : A n generators of Lie algebra ( Mde Guise, H.,et al PRA 2018) Elegant Mapping of the problem into a geometrical one: joint numerical range ( R. Schwonnek, et al., Phys. Rev. Lett. 119, 170404 (2017); K. Szyma ń ski and K. Ż yczkowski. arXiv:1804.06191 (2018).) Open problems: ARBITRARY NO OF OBSERVABLES, UNBOUNDED OBSERVABLES
Sum uncertainty relations L. Maccone, A. K. Pati, PRL 113 (26), 260401 (2014) Given N A n bounded operators acting on H M N ∆ 2 ∑ V Tot ( | ψ i ) = | ψ i A n � l B n = 1 the problem of trivial results is removed the goal is now to find a STATE INDEPENDENT LOWER BOUND l B direct numerical minimization HARD for large dimension M Di ff ernt strategies have been proposed Analyitc : A n generators of Lie algebra ( Mde Guise, H.,et al PRA 2018) Elegant Mapping of the problem into a geometrical one: joint numerical range ( R. Schwonnek, et al., Phys. Rev. Lett. 119, 170404 (2017); K. Szyma ń ski and K. Ż yczkowski. arXiv:1804.06191 (2018).) Open problems: ARBITRARY NO OF OBSERVABLES, UNBOUNDED OBSERVABLES
Outline Motivation 1 Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations First Result: Mapping onto eigenvalue problem 2 Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors Wrap up 3
Sum uncertainty relations: OUR SOLUTION Basic Mapping : For each operator A n define on H M ⌦ H M A 2 n ⌦ I + I ⌦ A 2 n H n = � A n ⌦ A n . (1) 2 then H Tot = ∑ n H n N ∑ V Tot ( | ψ i ) = h ψ | h ψ | H n | ψ i | ψ i � ε GS = l B (2) n = 1 state independent l B from ground state energy of H Tot = ∑ n H n Exact lowerbound Generators of su(2) ∆ 2 J X + ∆ 2 J Y + ∆ 2 J Z � j V XYZ = (3)
Sum uncertainty relations: OUR SOLUTION Basic Mapping : For each operator A n define on H M ⌦ H M A 2 n ⌦ I + I ⌦ A 2 n H n = � A n ⌦ A n . (1) 2 then H Tot = ∑ n H n N ∑ V Tot ( | ψ i ) = h ψ | h ψ | H n | ψ i | ψ i � ε GS = l B (2) n = 1 state independent l B from ground state energy of H Tot = ∑ n H n Exact lowerbound Generators of su(2) ∆ 2 J X + ∆ 2 J Y + ∆ 2 J Z � j V XYZ = (3)
Outline Motivation 1 Uncertainty Relations: Basic ideas and settings Sum Uncertainty relations First Result: Mapping onto eigenvalue problem 2 Basic Mapping Extension of the mapping State that approximately saturates the bound and Errors Wrap up 3
First Extension of the method for ε GS = 0 Ground state H Tot has a unique ground state that can be written in the eigenbasis of any a n , i i } M of the A n {| ˜ a n , i i | ˜ i = 1 as the maximally entangled state 1 M ∑ | ε gs i = p | ˜ a n , i i | ˜ a n , i i (4) i First exicted state energy bound: Given ε 1 on has ✓ 1 � 1 ◆ V Tot ( | ψ i ) ε 1 � M improvement but typically not very tight
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