Abstract This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. After an introduction in section 1, section 2 reviews the basic definitions and observables are considered in section 3. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in section 4. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, Lüders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author’s previous articles Gudder (2020 arXiv:2005.08117 [quant-ph], arXiv:2005.13642 [quant-ph], arXiv:2009.07371 [quant-ph]).Abstract This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. After an introduction in section 1, section 2 reviews the basic definitions and observables are considered in section 3. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in section 4. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, Lüders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author’s previous articles Gudder (2020 arXiv:2005.08117 [quant-ph], arXiv:2005.13642 [quant-ph], arXiv:2009.07371 [quant-ph]).

arXiv:2010.08025Journal of Physics A Mathematical and TheoreticalJul 27, 20218 citations

General Observables

Roderich Tumulka

Lecture notes in physicsJan 1, 20220 citations

Finite Quantum Instruments

Stanley Gudder

This article considers quantum systems described by a finite-dimensional complex Hilbert space $H$ . We first define the concept of a finite observable on $H$ . We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.This article considers quantum systems described by a finite-dimensional complex Hilbert space $H$. We first define the concept of a finite observable on $H$. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.

arXiv:2005.13642May 27, 20204 citations

Ramsey–CPT Resonance Observation Using Different Laser Sideband Combinations for the Two Interrogation Pulses

Masahiro Fukuoka, Shigeyoshi Goka

SummaryWe investigated the Ramsey–coherent population trapping (CPT) characteristics of a new observation method that employs different laser sideband combinations as the two interrogation pulses during Ramsey–CPT resonance observation. In an experimental setup using a gas cell containing Cs atoms, Ramsey–CPT resonance was observed using the 1st-1stsideband combination for the common preparation pulse and the 1st-1stand C-2nd sideband combinations for the detection pulse. The Ramsey–CPT spectral shapes were inverted to each other. The light shift characteristic was 2.7 times different to its light sensitivity. This difference in the light shift characteristics is expected to allow the estimation of the light frequency shift in the Ramsey–CPT atomic clock during operation.SummaryWe investigated the Ramsey–coherent population trapping (CPT) characteristics of a new observation method that employs different laser sideband combinations as the two interrogation pulses during Ramsey–CPT resonance observation. In an experimental setup using a gas cell containing Cs atoms, Ramsey–CPT resonance was observed using the 1st-1stsideband combination for the common preparation pulse and the 1st-1stand C-2nd sideband combinations for the detection pulse. The Ramsey–CPT spectral shapes were inverted to each other. The light shift characteristic was 2.7 times different to its light sensitivity. This difference in the light shift characteristics is expected to allow the estimation of the light frequency shift in the Ramsey–CPT atomic clock during operation.

Joint Conference of the European Frequency and Time Forum and IEEE International Frequency Control Symposium (EFTF/IFCS)Apr 24, 20222 citations

The Structure of Observables

Yoshio Ohnuki, S. Kamefuchi

Springer eBooksJan 1, 19820 citations

This article considers quantum systems described by a finite-dimensional complex Hilbert space

H

. We first define the concept of a finite observable on

H

. We then discuss ways of combining observables in terms of convex combinations, post-processing and sequential products. We also define complementary and coexistent observables. We then introduce finite instruments and their related compatible observables. The previous combinations and relations for observables are extended to instruments and their properties are compared. We present four types of instruments; namely, identity, trivial, Lüders and Kraus instruments. These types are used to illustrate different ways that instruments can act. We next consider joint probabilities for observables and instruments. The article concludes with a discussion of measurement models and the instruments they measure.