Entropy of Quantum Measurements

If <i>a</i> is a quantum effect and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a state, we define the <inline-formula...

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Bibliographic Details
Main Author: Stanley Gudder
Format: Article
Language:English
Published: MDPI AG 2022-11-01
Series:Entropy
Subjects:
Online Access:https://www.mdpi.com/1099-4300/24/11/1686
Description
Summary:If <i>a</i> is a quantum effect and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> is a state, we define the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> which gives the amount of uncertainty that a measurement of <i>a</i> provides about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>. The smaller <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is, the more information a measurement of <i>a</i> gives about <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>. In Entropy for Effects, we provide bounds on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and show that if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></semantics></math></inline-formula> is an effect, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mo>≥</mo><msub><mi>S</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>S</mi><mi>b</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We then prove a result concerning convex mixtures of effects. We also consider sequential products of effects and their <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-entropies. In Entropy of Observables and Instruments, we employ <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>a</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to define the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>A</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for an observable <i>A</i>. We show that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>A</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> directly provides the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi mathvariant="script">I</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for an instrument <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">I</mi></semantics></math></inline-formula>. We establish bounds for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>A</mi></msub><mrow><mo>(</mo><mi>ρ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and prove characterizations for when these bounds are obtained. These give simplified proofs of results given in the literature. We also consider <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-entropies for measurement models, sequential products of observables and coarse-graining of observables. Various examples that illustrate the theory are provided.
ISSN:1099-4300