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Overview of projective quantum measurements (2404.05679v1)

Published 8 Apr 2024 in quant-ph, cond-mat.quant-gas, and physics.atom-ph

Abstract: We provide an overview of standard "projective" quantum measurements with the goal of elucidating connections between theory and experiment. We make use of a unitary "Stinespring" representation of measurements on a dilated Hilbert space that includes both the physical degrees of freedom and those of the measurement apparatus. We explain how this unitary representation (i) is guaranteed by the axioms of quantum mechanics, (ii) relates to both the Kraus and von Neumann representations, and (iii) corresponds to the physical time evolution of the system and apparatus during the measurement process. The Stinespring representation also offers significant conceptual insight into measurements, helps connects theory and experiment, is particularly useful in describing protocols involving midcircuit measurements and outcome-dependent operations, and establishes that all quantum operations are compatible with relativistic locality, among other insights.

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References (88)
  1. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, U.K., 1930).
  2. G. Birkhoff and J. V. Neumann, The logic of quantum mechanics, Ann. Math. 37, 823 (1936).
  3. L. Hardy, Quantum theory from five reasonable axioms (2001), arXiv:quant-ph/0101012 [quant-ph] .
  4. C. A. Fuchs, Quantum mechanics as quantum information (and only a little more) (2002), arXiv:quant-ph/0205039 [quant-ph] .
  5. G. Mackey, Mathematical Foundations of Quantum Mechanics (Dover, 2004).
  6. A. Wilce, Four and a half axioms for finite-dimensional quantum mechanics (2009), arXiv:0912.5530 [quant-ph] .
  7. L. Masanes and M. P. Müller, A derivation of quantum theory from physical requirements, New J. Phys. 13, 063001 (2011).
  8. A. Kapustin, Is quantum mechanics exact?, J. Math. Phys. 54, 062107 (2013).
  9. A. Einstein, Letter to Max Born, 1926, in The Born-Einstein Letters (Walker and Company, New York, 1971) translated by Irene Born.
  10. A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Phys. Rev. 47, 777 (1935).
  11. J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurement, Vol. 53 (Princeton University Press, US, 2014).
  12. H. Margenau, Measurements in quantum mechanics, Ann. Phys. 23, 469 (1963).
  13. A. Peres, When is a quantum measurement?, Amer. J. Phys. 54, 688 (1986).
  14. G. Lindblad, Entropy, information and quantum measurements, Commun. Math. Phys. 33, 305 (1973).
  15. M. Schlosshauer, Decoherence and the Quantum-To-Classical Transition, Frontiers Collection (Springer Berlin, Heidelberg, Germany, 2007).
  16. C. A. Brasil, F. F. Fanchini, and R. d. J. Napolitano, A simple derivation of the Lindblad equation, Revista Brasileira de Ensino de Física 35, 01 (2013).
  17. K.-E. Hellwig and K. Kraus, Pure operations and measurements, Commun. Math. Phys. 11, 214 (1969).
  18. K. Kraus, General state changes in quantum theory, Ann. Phys. 64, 311 (1971).
  19. K. Kraus, Measuring processes in quantum mechanics I. Continuous observation and the watchdog effect, Found. Phys. 11, 547 (1981).
  20. A. J. Friedman, O. Hart, and R. Nandkishore, Measurement-induced phases of matter require feedback (2022b), arXiv:2210.07256 [quant-ph] .
  21. Y. Hong, D. T. Stephen, and A. J. Friedman, Quantum teleportation implies symmetry-protected topological order (2023), arXiv:2310.12227 [quant-ph] .
  22. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, UK, 2010).
  23. R. Jozsa, An introduction to measurement-based quantum computation (2005), arXiv:quant-ph/0508124 [quant-ph] .
  24. V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, UK, 2003).
  25. M. Takesaki, Theory of Operator Algebras I (Springer New York, New York, US, 2012).
  26. W. F. Stinespring, Positive functions on C*superscript𝐶{C}^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras, Proc. Amer. Math. Soc. 6, 211 (1955).
  27. M.-D. Choi, Completely positive linear maps on complex matrices, Lin. Alg. Appl. 10, 285 (1975).
  28. M. M. Wolf, Quantum Channels and Operations — Guided Tour (2012), graue Literatur.
  29. A. J. Friedman and J. Woodcock, Dilation theorems for quantum operations (2024), to appear.
  30. J. von Neumann, Mathematical Foundations of Quantum Mechanics: New Edition, edited by R. T. Beyer and N. A. Wheeler (Princeton University Press, US, 2018).
  31. J. Preskill, The Physics of Quantum Information (2022), arXiv:2208.08064 [quant-ph] .
  32. K. R. Davidson, C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-Algebras by Example, in Fields Institute Monographs, Vol. 6 (AMS, 1996).
  33. O. Bratteli, Inductive limits of finite-dimensional C*superscript𝐶C^{*}italic_C start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT-algebras, Trans. Amer. Math. Soc. 171, 195 (1972).
  34. P. D. Drummond and Z. Ficek, Quantum Squeezing (Springer Berlin, Heidelberg, Germany, 2004).
  35. K. Jacobs, Quantum Measurement Theory and its Applications (Cambridge University Press, 2014).
  36. W. Gerlach and O. Stern, Der experimentelle Nachweis des magnetischen Moments des Silberatoms, Zeitschrift für Physik 8, 110 (1922a).
  37. W. Gerlach and O. Stern, Das magnetische Moment des Silberatoms, Zeitschrift für Physik 9, 353 (1922b).
  38. W. Gerlach and O. Stern, Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld, Zeitschrift für Physik 9, 349 (1922c).
  39. P. Busch, Translation of “Die Messung quantenmechanischer Operatoren” by E.P. Wigner (2010), arXiv:1012.4372 [quant-ph] .
  40. H. Araki and M. M. Yanase, Measurement of quantum mechanical operators, Phys. Rev. 120, 622 (1960).
  41. M. Le Bellac, Quantum Physics, edited by P. Forcrand-Millard (Cambridge University Press, 2006).
  42. P. Ehrenfest, Bemerkung über die angenäherte Gültigkeit der klassischen Mechanik innerhalb der Quantenmechanik, Zeitschrift für Physik 45, 455 (1927).
  43. W. Magnus, On the exponential solution of differential equations for a linear operator, CPAM 7, 649 (1954).
  44. A. Reiserer, S. Ritter, and G. Rempe, Nondestructive detection of an optical photon, Science 342, 1349 (2013).
  45. R. J. Glauber, The quantum theory of optical coherence, Phys. Rev. 130, 2529 (1963).
  46. J. Combes and A. P. Lund, Homodyne measurement with a Schrödinger cat state as a local oscillator, Phys. Rev. A 106, 063706 (2022).
  47. H. P. Yuen and J. H. Shapiro, Quantum statistics of homodyne and heterodyne detection, in Coherence and Quantum Optics IV, edited by L. Mandel and E. Wolf (Springer, Boston, US, 1978).
  48. M. J. Collett, R. Loudon, and C. Gardiner, Quantum theory of optical homodyne and heterodyne detection, J. Mod. Opt. 34, 881 (1987).
  49. A. Barchielli, Direct and heterodyne detection and other applications of quantum stochastic calculus to quantum optics, J. Eur. Opt. Soc. B Quant. Opt. 2, 423 (1990).
  50. H. M. Wiseman and G. J. Milburn, Quantum theory of field-quadrature measurements, Phys. Rev. A 47, 642 (1993).
  51. A. I. Lvovsky and M. G. Raymer, Continuous-variable optical quantum-state tomography, Rev. Mod. Phys. 81, 299 (2009).
  52. P. N. J. Dennis, Photodetectors: An Introduction to Current Technology (Springer US, 1986).
  53. B. R. Mollow, Quantum theory of field attenuation, Phys. Rev. 168, 1896 (1968).
  54. M. O. Scully and W. E. Lamb, Quantum theory of an optical maser. III. Theory of photoelectron counting statistics, Phys. Rev. 179, 368 (1969).
  55. M. Srinivas and E. Davies, Photon counting probabilities in quantum optics, Optica Acta: Int. J. Opt. 28, 981 (1981).
  56. W. Chmara, A quantum open-systems theory approach to photodetection, J. Mod. Opt. 34, 455 (1987).
  57. M. Fleischhauer and D. G. Welsch, Nonperturbative approach to multimode photodetection, Phys. Rev. A 44, 747 (1991).
  58. C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed., Springer Series in Synergetics (Springer, 2004).
  59. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, UK, 2020).
  60. T. Tyc and B. C. Sanders, Operational formulation of homodyne detection, J. Phys. A Math. Gen. 37, 7341 (2004).
  61. B. Yurke, S. L. McCall, and J. R. Klauder, SU(2) and SU(1,1) interferometers, Phys. Rev. A 33, 4033 (1986).
  62. S. Prasad, M. O. Scully, and W. Martienssen, A quantum description of the beam splitter, Opt. Commun. 62, 139 (1987).
  63. S. Wallentowitz and W. Vogel, Unbalanced homodyning for quantum state measurements, Phys. Rev. A 53, 4528 (1996).
  64. A. Cives-Esclop, A. Luis, and L. Sánchez-Soto, Unbalanced homodyne detection with a weak local oscillator, Optics Communications 175, 153 (2000).
  65. B. Kühn and W. Vogel, Unbalanced homodyne correlation measurements, Phys. Rev. Lett. 116, 163603 (2016).
  66. J. H. Shapiro, The quantum theory of optical communications, IEEE J. Quant. Electr. 15, 1547 (2009).
  67. I. Bloch, J. Dalibard, and S. Nascimbène, Quantum simulations with ultracold quantum gases, Nature Phys. 8, 267 (2012).
  68. H. Ott, Single-atom detection in ultracold quantum gases: A review of current progress, Rep. Prog. Phys. 79, 054401 (2016).
  69. W. Nagourney, J. Sandberg, and H. Dehmelt, Shelved optical electron amplifier: Observation of quantum jumps, Phys. Rev. Lett. 56, 2797 (1986).
  70. E. Jaynes and F. Cummings, Comparison of quantum and semiclassical radiation theories with application to the beam maser, Proceedings of the IEEE 51, 89 (1963).
  71. M. H. Schleier-Smith, I. D. Leroux, and V. Vuletić, States of an ensemble of two-level atoms with reduced quantum uncertainty, Phys. Rev. Lett. 104, 073604 (2010).
  72. C. C. Bradley, C. A. Sackett, and R. G. Hulet, Bose-Einstein condensation of lithium: Observation of limited condensate number, Phys. Rev. Lett. 78, 985 (1997).
  73. A. Gleason, Measures on the closed subspaces of a Hilbert space, Indiana Univ. Math. J. 6, 885 (1957).
  74. E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, Commun. Math. Phys. 28, 251 (1972).
  75. D. Poulin, Lieb-Robinson bound and locality for general Markovian quantum dynamics, Phys. Rev. Lett. 104, 190401 (2010).
  76. J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  77. M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  78. M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008).
  79. B. Doyon, Thermalization and pseudolocality in extended quantum systems, Commun. Math. Phys. 351, 155 (2017).
  80. O. Bohigas, M. J. Giannoni, and C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws, Phys. Rev. Lett. 52, 1 (1984).
  81. D. A. Roberts and B. Yoshida, Chaos and complexity by design, JHEP 2017 (4).
  82. A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits (2017), arXiv:1705.08975 [cond-mat.str-el] .
  83. T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation (2017), arXiv:1710.09827 [cond-mat.stat-mech] .
  84. J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics Physique Fizika 1, 195 (1964).
  85. M. B. Hastings, Locality in quantum systems (2010), arXiv:1008.5137 [math-ph] .
  86. S. Popescu and D. Rohrlich, Quantum nonlocality as an axiom, Found. Phys. 24, 379 (1994).
  87. C.-F. Chen and A. Lucas, Finite speed of quantum scrambling with long-range interactions, Phys. Rev. Lett. 123, 250605 (2019).
  88. C.-F. Chen, A. Lucas, and C. Yin, Speed limits and locality in many-body quantum dynamics, Rep. Prog. Phys. 86, 116001 (2023).
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