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Question 20: Quantum Probability and Spectral Measures Problem Statement: In quantum probability, observables are represented by self-adjoint operators on a Hilbert space H. Let X be a self-adjoint operator representing a quantum observable with spectral measure E. 1. Probability Measures from Quantum States: Given a quantum state p (a positive trace-class operator with trace one), define the probability measure μp on R by μ(B) = Tr(pE(B)) for Borel sets B. Prove that up is a well-defined probability measure. 2. Measure-Theoretic Expectation and Variance: Show that the expectation E(X) and variance Var, (X) of the observable X in the state p can be expressed as E(X) = [Adµ(A), Var(x) = [ {A – E‚(X))² dµ‚(A). Provide detailed proofs of these expressions using measure-theoretic integration. 3. Spectral Measures and Quantum Measurements: Analyze how the spectral measure E encodes the probabilities of different measurement outcomes for X and prove that successive measurements correspond to measure-theoretic operations on the spectral measures associated with X. Requirements: Connect spectral theory with the foundations of quantum probability.

Question 20: Quantum Probability and Spectral Measures Problem Statement: In quantum probability, observables are represented by self-adjoint operators on a Hilbert space H. Let X be a self-adjoint operator representing a quantum observable with spectral measure E. 1. Probability Measures from Quantum States: Given a quantum state p (a positive trace-class operator with trace one), define the probability measure μp on R by μ(B) = Tr(pE(B)) for Borel sets B. Prove that up is a well-defined probability measure. 2. Measure-Theoretic Expectation and Variance: Show that the expectation E(X) and variance Var, (X) of the observable X in the state p can be expressed as E(X) = [Adµ(A), Var(x) = [ {A – E‚(X))² dµ‚(A). Provide detailed proofs of these expressions using measure-theoretic integration. 3. Spectral Measures and Quantum Measurements: Analyze how the spectral measure E encodes the probabilities of different measurement outcomes for X and prove that successive measurements correspond to measure-theoretic operations on the spectral measures associated with X. Requirements: Connect spectral theory with the foundations of quantum probability.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter2: Matrices
Section2.5: Markov Chain
Problem 34E
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Question 20: Quantum Probability and Spectral Measures Problem Statement: In quantum probability, observables are represented by self-adjoint operators on a Hilbert space H. Let X be a self-adjoint operator representing a quantum observable with spectral measure E. 1. Probability Measures from Quantum States: Given a quantum state p (a positive trace-class operator with trace one), define the probability measure μp on R by μ(B) = Tr(pE(B)) for Borel sets B. Prove that up is a well-defined probability measure. 2. Measure-Theoretic Expectation and Variance: Show that the expectation E(X) and variance Var, (X) of the observable X in the state p can be expressed as E(X) = [Adµ(A), Var(x) = [ {A – E‚(X))² dµ‚(A). Provide detailed proofs of these expressions using measure-theoretic integration. 3. Spectral Measures and Quantum Measurements: Analyze how the spectral measure E encodes the probabilities of different measurement outcomes for X and prove that successive measurements correspond to measure-theoretic operations on the spectral measures associated with X. Requirements: Connect spectral theory with the foundations of quantum probability.
Transcribed Image Text:Question 20: Quantum Probability and Spectral Measures Problem Statement: In quantum probability, observables are represented by self-adjoint operators on a Hilbert space H. Let X be a self-adjoint operator representing a quantum observable with spectral measure E. 1. Probability Measures from Quantum States: Given a quantum state p (a positive trace-class operator with trace one), define the probability measure μp on R by μ(B) = Tr(pE(B)) for Borel sets B. Prove that up is a well-defined probability measure. 2. Measure-Theoretic Expectation and Variance: Show that the expectation E(X) and variance Var, (X) of the observable X in the state p can be expressed as E(X) = [Adµ(A), Var(x) = [ {A – E‚(X))² dµ‚(A). Provide detailed proofs of these expressions using measure-theoretic integration. 3. Spectral Measures and Quantum Measurements: Analyze how the spectral measure E encodes the probabilities of different measurement outcomes for X and prove that successive measurements correspond to measure-theoretic operations on the spectral measures associated with X. Requirements: Connect spectral theory with the foundations of quantum probability.
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