Abstract
We analyze general enough models of repeated indirect measurements in which a quantum system interacts repeatedly with randomly chosen probes on which von Neumann direct measurements are performed. We prove, under suitable hypotheses, that the system state probability distribution converges after a large number of repeated indirect measurements, in a way compatible with quantum wave function collapse. We extend this result to mixed states and we prove similar results for the system density matrix. We show that the convergence is exponential with a rate given by some relevant mean relative entropies. We also prove that, under appropriate rescaling of the system and probe interactions, the state probability distribution and the system density matrix are solutions of stochastic differential equations modeling continuous-time quantum measurements. We analyze the large time behavior of these continuous time processes and prove convergence.
Article PDF
Similar content being viewed by others
References
Wheeler, J.A., Zurek, W.H. (eds): Quantum Theory and Measurements. Princeton University Press, Princeton (1983)
Guerlin C. et al.: Progressive field-state collapse and quantum non-demolition photon counting. Nature 448, 889 (2007)
Devoret M., Martinis J.M.: Implementing qubits with superconducting integrated circuits. Quantum Inf. Process. 3, 163–203 (2004)
Bauer M., Bernard D.: Convergence of repeated quantum nondemolition measurements and wave-function collapse. Phys. Rev. A 84, 044103 (2011)
Kummerer B., Maassen H.: A pathwise ergodic theorem for quantum trajectories. J. Phys. A 37(49), 11889–11896 (2004)
Maassen, H., Kummerer, B.: Purification of quantum trajectories in dynamics and stochastics. In: IMS Lecture Notes Monogr. Ser., vol. 48, pp. 252–261. Inst. Math. Statist., Beachwood, OH (2006)
Bouten L., van Handel R., James M.R.: A discrete invitation to quantum filtering and feedback control. SIAM Rev. 51(2), 239–316 (2009)
Davies E.B.: Quantum Theory of Open Systems. Academic, New York (1976)
Gisin N.: Quantum measurements and stochastic processes. Phys. Rev. Lett. 52, 1657–1660 (1984)
Diosi L.: Quantum stochastic processes as models for quantum state reduction. J. Phys. A21, 2885 (1988)
Barchielli A., Belavkin V.P.: Measurements continuous in time and a posteriori states in quantum mechanics. J. Phys. A Math. Gen. 24, 1495–1514 (1991)
Barchielli A.: Measurement theory and stochastic differential equations in quantum mechanics. Phys. Rev. A 34, 1642–1648 (1986)
Belavkin V.P.: A new wave equation for a continuous nondemolition measurement. Phys. Lett. A 140, 355–358 (1989)
Belavkin V.P.: A posterior Schrödinger equation for continuous non demolition measurement. J. Math. Phys. 31, 2930–2934 (1990)
Wiseman H.M.: Quantum theory and continuous feedback. Phys. Rev. A49, 2133 (1994)
Bouten L., Guţă M., Maassen H.: Stochastic Schrödinger equations. J. Phys. A Math. Gen. 37, 3189–3209 (2004)
Bouten L., van Handel R., James M.R.: An introduction to quantum filtering. SIAM J. Control Optim. 46, 2199 (2007)
Pellegrini C.: Markov chains approximation of jump-diffusion stochastic master equations. Ann. Henri Poincaré 46, 924–948 (2010)
Pellegrini C.: Existence, uniqueness and approximation for stochastic Schrödinger equation: the diffusive case. Ann. Probab. 36, 2332–2353 (2008)
Pellegrini C.: Existence, uniqueness and approximation of the jump-type stochastic Schrödinger equation for two-level systems. Stoch. Proc. Appl. 120, 1722–1747 (2010)
Attal S., Pautrat Y.: From repeated to continuous quantum interactions. Ann. Henri Poincaré 7(1), 59104 (2006)
Belavkin V.P.: Quantum stochastic calculus and quantum nonlinear filtering. J. Multivar. Anal. 42, 171–201 (1992)
Belavkin V.P.: Quantum continual measurements and a posteriori collapse on CCR. Commun. Math. Phys. 146, 611–635 (1992)
Adler S.L. et al.: Martingale models for quantum state reduction. J. Phys. A 34, 8795 (2001)
van Handel R., Stockton J., Mabuchi H.: Feedback control of quantum state reduction. IEEE Trans. Autom. Control 50, 768 (2005)
Stockton J., van Handel R., Mabuchi H.: Deterministic Dick-state preparation with continuous measurement and control. Phys. Rev. A70, 022106 (2004)
Amini, H., Mirrahimi, M., Rouchon, P.: Design of Strict control-Lyapunov functions for quantum systems with QND measurements. CDC/ECC 2011. http://arxiv.org/abs/1103.1365
Øksendal B.K.: Stochastic differential equations: an introduction with applications. Springer, Berlin (2003)
Rouchon P.: Fidelity is a sub-martingale for discrete-time quantum filters. IEEE Trans. Autom. Control 56(11), 2743–2747 (2011)
Pellegrini, C., Benoist, T.: (in preparation)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Krzysztof Gawedzki.
D. Bernard: Member of CNRS.
Rights and permissions
About this article
Cite this article
Bauer, M., Benoist, T. & Bernard, D. Repeated Quantum Non-Demolition Measurements: Convergence and Continuous Time Limit. Ann. Henri Poincaré 14, 639–679 (2013). https://doi.org/10.1007/s00023-012-0204-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-012-0204-x