Quantum Measurement Theory on a Half Line

1. PS-09 Quantum Measurement Theory on a Half Line Yutaka Shikano Department of Physics, Tokyo Institute of Technology Collaborator: Akio Hosoya Y. Shikano and A. Hosoya, in preparation

2. Outline and Aim • What is Quantum Information? • What is Measurement? (e.g. Measuring Process) • What is Covariant Measurement? • Comments on the Momentum on a Half Line • Optimal Covariant Measurement Model Why need we consider Quantum Measurement? Why need we consider the Half Line system? （Details: Y. Shikano and A. Hosoya, in preparation） The 52nd Condensed Matter Physics Summer School at Wakayama

3. Preparation Measurement Initial Conditions Output Data Object What is Quantum Information? • Operational Processes in the Quantum System My Research Field Similar to Information Process Solve the Schroedinger Equations. What information do we obtain from this result? The 52nd Condensed Matter Physics Summer School at Wakayama

4. Aim of Quantum Information • Solve the Schroedinger Equation. • Understand how to obtain Information from Quantum System. • How much information can we get? • What method can we obtain information optimally? • Question: Is the essence of quantum mechanics the operational concept? The 52nd Condensed Matter Physics Summer School at Wakayama

5. Axioms of Quantum Mechanics • Definition of state, state space, observable • Observable is defined as the self-adjoint operator since the operator has real spectrums. • Time evolution of state (Schroedinger Equation) • Born’s probablistic formula • Definition of the combined system The 52nd Condensed Matter Physics Summer School at Wakayama

6. 1 2 3 What is Measurement? Measured System Probe System t = 0 Interaction between the measured system and probe system. We can evaluate the “measurement” value t = 0 on the measured system from the measurement value t = ⊿t. We obtain the measurement value on the probe system. t = ⊿t This process is called magnification or observation and is different from measurement. t = ⊿t+⊿T We obtain the macroscopic value. (e.g. Photomultiplier) time The 52nd Condensed Matter Physics Summer School at Wakayama

7. To describe the Measuring Process • We have to know the follows to describe the measuring process physically. • Hamiltonian on the combined system between the measured system and probe system. • Evolution operator on the combined system from the Hamiltonian • Measuring time of the measuring process. • Measurement value of the probe system. The 52nd Condensed Matter Physics Summer School at Wakayama

8. 1 2 0 3 0 3 measurement value measurement value 3 0 What is Covariant Measurement? For any bases “0” on the space, Measured System This condition is satisfied by the ideal measuring device. Shifted Probe System Remark The measure on the measured system, that is POVM, is constrained. Shifted as same!! The 52nd Condensed Matter Physics Summer School at Wakayama

9. Formulation of the Covariant Measurement Definition transformation of the momentum. Property of the covariant measurement. Using the Born formula. (Axiom 3) (Holevo 1978,1979,1982) The 52nd Condensed Matter Physics Summer School at Wakayama

10. Comments on the Momentum on a Half Line NOT SAME The 52nd Condensed Matter Physics Summer School at Wakayama

11. Lesson from this example. • is symmetric but not self-adjoint operator. This means that the momentum on a half line is NOT observable. • Lesson: • When we consider the infinite dimensional Hilbert space, e.g. momentum and position in quantum mechanics, we have to check the domainof the operator. How to classify the operator. (Weyl 1910, von Neumann 1929, Bonneau et al. 2001) Deficiency Theorem The 52nd Condensed Matter Physics Summer School at Wakayama

12. 3 3 2 1 position Combined position 0 0 Copy Inversion position position 0 0 Prescription: Naimark Extension • Naimark Extension Theorem: • When we extend the domain of any symmetric operators, the symmetric operators become the self-adjoint operator on the extended domain. The 52nd Condensed Matter Physics Summer School at Wakayama

13. Optimal Covariant Measurement Model • Aim: • We find the Hamiltonian to satisfy the optimal covariant POVM to minimize the variance between the measurement value on the probe system at t = ⊿t and the evaluated “measurement” value on the measured system at t = 0. Optimal Covariant POVM The 52nd Condensed Matter Physics Summer School at Wakayama

14. Model Hamiltonian Assume that the measured system alone is coupled to the bulk system at zero temperature. Evolution operator The 52nd Condensed Matter Physics Summer School at Wakayama

15. Remarks • Following iεprescription, we can obtain the optimal covariant POVM. Measured System Probe System Bulk System Instantaneous interaction T=0 t = 0 Energy Dissipates! Precise evaluation from the momentum conservation. Controllable T=0 Ground State t = ∞ time Assumption: Ground Energy = 0 Measurable The 52nd Condensed Matter Physics Summer School at Wakayama

16. Concluding Remarks • We have explained the overview of quantum information and the measurement theory of quantum system. • We have shown the strange example of the half line system. • We have obtained the optimal covariant measurement model. Thank you for your attention although my poster presentation may be out of place in this session. The 52nd Condensed Matter Physics Summer School at Wakayama

17. References • Y. Shikano and A. Hosoya, in preparation • J. von Neumann, Mathematische Grundlagen der Quantmechanik (Springer, Berlin, 1932), [ Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, 1955) ] • A. S. Holevo, Rep. Math. Phys. 13, 379-399 (1978) • A. S. Holevo, Rep. Math. Phys. 16, 385-400 (1979) • A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982) • H. Weyl, Math. Ann. 68, 220-269 (1910) • J. von Neumann, Math. Ann. 102, 49-131 (1929) • G. Bonneau et al. Am. J. Phys. 69, 322-331 (2001) The 52nd Condensed Matter Physics Summer School at Wakayama