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Unsharp Measurements and Joint Measurability


Affiliations
1 Raman Research Institute, Bengaluru 560 080, India
2 Department of Physics, Bangalore University, Bengaluru 560 056, India and Inspire Institute Inc., Alexandria, Virginia 22303, United States
3 Inspire Institute Inc., Alexandria, Virginia, 22303, USA; Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, and Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
 

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix - which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables - could be realized if and only if compatible POVMs are employed.

Keywords

Incompatibility, Joint Measurability, Positive Operator Valued Measures, Unsharp Measurements.
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  • Unsharp Measurements and Joint Measurability

Abstract Views: 377  |  PDF Views: 139

Authors

H. S. Karthik
Raman Research Institute, Bengaluru 560 080, India
A. R. Usha Devi
Department of Physics, Bangalore University, Bengaluru 560 056, India and Inspire Institute Inc., Alexandria, Virginia 22303, United States
A. K. Rajagopal
Inspire Institute Inc., Alexandria, Virginia, 22303, USA; Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, and Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract


We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix - which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables - could be realized if and only if compatible POVMs are employed.

Keywords


Incompatibility, Joint Measurability, Positive Operator Valued Measures, Unsharp Measurements.

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DOI: https://doi.org/10.18520/cs%2Fv109%2Fi11%2F2061-2068