Skip to main content

News about Scitation

In December 2016 Scitation will launch with a new design, enhanced navigation and a much improved user experience.

To ensure a smooth transition, from today, we are temporarily stopping new account registration and single article purchases. If you already have an account you can continue to use the site as normal.

For help or more information please visit our FAQs.

banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
1.R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed. (Springer, Berlin, 1996).
2.J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932).
3.M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000).
4.V. B. Braginsky, Y. I. Vorontsov, and K. S. Thorne, “Quantum nondemolition measurements,” Science 209, 547557 (1980).
5.H. P. Yuen, “Contractive states and the standard quantum limit for monitoring free-mass positions,” Phys. Rev. Lett. 51, 719722 (1983).
6.C. M. Caves, “Defense of the standard quantum limit for free-mass position,” Phys. Rev. Lett. 54, 24652468 (1985).
7.M. Ozawa, “Measurement breaking the standard quantum limit for free-mass position,” Phys. Rev. Lett. 60, 385388 (1988).
8.M. Ozawa, “Realization of measurement and the standard quantum limit,” in Squeezed and Nonclassical Light, edited by P. Tombesi and E. R. Pike (Plenum, New York, 1989), pp. 263286 ; e-print arXiv:1505.01083 [quant-ph].
9.J. Maddox, “Beating the quantum limits,” Nature 331, 559 (1988).
10.W. Heisenberg, “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” Z. Phys. 43, 172198 (1927).
11.D. M. Appleby, “The error principle,” Int. J. Theor. Phys. 37, 25572572 (1998).
12.M. Ozawa, “Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement,” Phys. Rev. A 67, 042105 (2003).
13.M. Ozawa, “Physical content of Heisenberg’s uncertainty relation: Limitation and reformulation,” Phys. Lett. A 318, 2129 (2003).
14.M. Ozawa, “Uncertainty principle for quantum instruments and computing,” Int. J. Quantum Inf. 1, 569588 (2003).
15.M. Ozawa, “Uncertainty relations for joint measurements of noncommuting observables,” Phys. Lett. A 320, 367374 (2004).
16.M. Ozawa, “Uncertainty relations for noise and disturbance in generalized quantum measurements,” Ann. Phys. 311, 350416 (2004).
17.M. J. W. Hall, “Prior information: How to circumvent the standard joint-measurement uncertainty relation,” Phys. Rev. A 69, 052113 (2004).
18.R. F. Werner, “The uncertainty relation for joint measurement of position and momentum,” Quantum Inf. Comput. 4, 546562 (2004).
19.C. Branciard, “Error-tradeoff and error-disturbance relations for incompatible quantum measurements,” Proc. Natl. Acad. Sci. U. S. A. 110, 67426747 (2013).
20.P. Busch, P. Lahti, and R. F. Werner, “Proof of Heisenberg’s error-disturbance relation,” Phys. Rev. Lett. 111, 160405 (2013).
21.F. Buscemi, M. J. W. Hall, M. Ozawa, and M. M. Wilde, “Noise and disturbance in quantum measurements: An information-theoretic approach,” Phys. Rev. Lett. 112, 050401 (2014).
22.P. Busch, P. Lahti, and R. F. Werner, “Measurement uncertainty relations,” J. Math. Phys. 55, 042111 (2014).
23.C. Branciard, “Deriving tight error-trade-off relations for approximate joint measurements of incompatible quantum observables,” Phys. Rev. A 89, 022124 (2014).
24.J. Erhart, S. Sponar, G. Sulyok, G. Badurek, M. Ozawa, and Y. Hasegawa, “Experimental demonstration of a universally valid error-disturbance uncertainty relation in spin measurements,” Nat. Phys. 8, 185189 (2012).
25.L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
26.S.-Y. Baek, F. Kaneda, M. Ozawa, and K. Edamatsu, “Experimental violation and reformulation of the Heisenberg error-disturbance uncertainty relation,” Sci. Rep. 3, 2221 (2013).
27.G. Sulyok, S. Sponar, J. Erhart, G. Badurek, M. Ozawa, and Y. Hasegawa, “Violation of Heisenberg’s error-disturbance uncertainty relation in neutron spin measurements,” Phys. Rev. A 88, 022110 (2013).
28.M. Ringbauer, D. N. Biggerstaff, M. A. Broome, A. Fedrizzi, C. Branciard, and A. G. White, “Experimental joint quantum measurements with minimum uncertainty,” Phys. Rev. Lett. 112, 020401 (2014).
29.F. Kaneda, S.-Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
30.G. Sulyok, S. Sponar, B. Demirel, F. Buscemi, M. J. W. Hall, M. Ozawa, and Y. Hasegawa, “Experimental test of entropic noise-disturbance uncertainty relations for spin-1/2 measurements,” Phys. Rev. Lett. 115, 030401 (2015).
31.M. Nakamura and H. Umegaki, “On von Neumann’s theory of measurements in quantum statistics,” Math. Japon. 7, 151157 (1962).
32.H. Umegaki, “Conditional expectation in an operator algebra,” Tôhoku Math. J. 6(2), 177181 (1954).
33.W. Arveson, “Analyticity in operator algebras,” Am. J. Math. 89, 578642 (1967).
34.E. B. Davies and J. T. Lewis, “An operational approach to quantum probability,” Commun. Math. Phys. 17, 239260 (1970).
35.E. B. Davies, Quantum Theory of Open Systems (Academic, London, 1976).
36.J. Schwinger, “The algebra of microscopic measurement,” Proc. Natl. Acad. Sci. U. S. A. 45, 15421554 (1959);
36.J. Schwinger, “The geometry of quantum states,” Proc. Natl. Acad. Sci. U. S. A. 46, 257265 (1960).
37.R. Haag and D. Kastler, “An algebraic approach to quantum field theory,” J. Math. Phys. 5, 848861 (1964).
38.G. Ludwig, “Attempt of an axiomatic foundation of quantum mechanics and more general theories, II,” Commun. Math. Phys. 4, 331348 (1967);
38.G. Ludwig, “Attempt of an axiomatic foundation of quantum mechanics and more general theories, III,” Commun. Math. Phys. 9, 112 (1968).
39.K. Kraus, “General state changes in quantum theory,” Ann. Phys. 64, 311335 (1971).
40.K. Kraus, States, Effects, and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics (Springer, Berlin, 1983), Vol. 190.
41.M. Ozawa, “Quantum measuring processes of continuous observables,” J. Math. Phys. 25, 7987 (1984).
42.W. Arveson, An Invitation to C*-Algebras (Springer, New York, 1976).
43.H. Araki, Mathematical Theory of Quantum Fields (Oxford University Press, Oxford, 2000).
44.M. Ozawa, “Conditional probability and a posteriori states in quantum mechanics,” Publ. Res. Inst. Math. Sci. 21, 279295 (1985).
45.S. Doplicher, R. Haag, and J. Roberts, “Fields, observables and gauge transformations, I,” Commun. Math. Phys. 13, 123 (1969);
45.S. Doplicher, R. Haag, and J. Roberts, “Fields, observables and gauge transformations, II,” Commun. Math. Phys. 15, 173200 (1969);
45.S. Doplicher, R. Haag, and J. Roberts, “Local observables and particle statistics, I,” Commun. Math. Phys 23, 199230 (1971);
45.S. Doplicher, R. Haag, and J. Roberts, “Local observables and particle statistics, II,” Commun. Math. Phys 35, 4985 (1974).
46.S. Doplicher and J. Roberts, “Endomorphism of C*-algebras, cross products and duality for compact groups,” Ann. Math. 130, 75119 (1989);
46.S. Doplicher and J. Roberts, “A new duality theory for compact groups,” Invent. Math. 98, 157218 (1989);
46.S. Doplicher and J. Roberts, “Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics,” Commun. Math. Phys. 131, 51107 (1990).
47.D. Buchholz and E. Wichmann, “Causal independence and the energy-level density of states in local quantum field theory,” Commun. Math. Phys. 106, 321344 (1986).
48.M. Takesaki, Theory of Operator Algebras I (Springer, New York, 1979).
49.M. Takesaki, Theory of Operator Algebras III (Springer, Berlin, 2002).
50.E. Effros and E. Lance, “Tensor products of operator algebras,” Adv. Math. 25, 134 (1977).
51.N. Brown and N. Ozawa, C*-Algebras and Finite-Dimensional Approximations (American Mathematical Society, Providence, 2008).
52.W. Arveson, “Subalgebras of C*-algebras,” Acta Math. 123, 141224 (1969).
53.V. Paulsen, Completely Bounded Maps and Operator Algebras (Cambridge University Press, Cambridge, 2002).
54.M. Ozawa, “Concepts of conditional expectations in quantum theory,” J. Math. Phys. 26, 19481955 (1985).
55.G. W. Mackey, “Borel structure in groups and their duals,” Trans. Am. Math. Soc. 85, 134165 (1957).
56.I. Ojima, K. Okamura, and H. Saigo, “Local state and sector theory in local quantum physics,” e-print arXiv:1501.00234 [math-ph] (2015).
57.J. Dixmier, Von Neumann Algebras (North-Holland, Amsterdam, 1981).
58.E. B. Davies, “On the repeated measurement of continuous observables in quantum mechanics,” J. Funct. Anal. 6, 318346 (1970).
59.M. Ozawa, “An operational approach to quantum state reduction,” Ann. Phys. 259, 121137 (1997).
60.M. Ozawa, “Mathematical foundations of quantum information: Measurement and foundations,” Sugaku Expositions 27, 195221 (2014); e-print arXiv:1201.5334 [quant-ph].
61.W. Arveson, “The noncommutative Choquet boundary,” J. Am. Math. Soc. 21, 10651084 (2008).
62.M. Raginsky, “Radon-Nikodym derivatives of quantum operations,” J. Math. Phys. 44, 50035020 (2003).
63.C. Anantharaman-Delaroche, “Amenable correspondences and approximation properties for von Neumann algebras,” Pac. J. Math. 171, 309341 (1995).
64.D. Buchholz, C. D’Antoni, and K. Fredenhagen, “The universal structure of local algebras,” Commun. Math. Phys. 111, 123135 (1987).
65.A. Connes, Noncommutative Geometry (Academic, San Diego, CA, 1994).
66.A. Grothendieck, Produits Tensoriels Topologiques et Espaces Nucléaires, Memoirs of the American Mathematical Society (American Mathematical Society, Providence, RI, 1955), Vol. 16.
67.S. Sakai, C*-Algebras and W*-Algebras (Springer, Berlin, 1971).
68.A. I. Tulcea and C. I. Tulcea, Topics in the Theory of Lifting (Springer, Heidelberg, 1969).
69.P. R. Halmos, Introduction to Hilbert Space and the Theory of Spectral Multiplicity (Chelsea, New York, 1951).
70.A. Kolmogorov and S. Fomin, Elements of the Theory of Functions and Functional Analysis (Academic, New York, 1961).
71.G. Lüders, “Über die Zustandsänderung durch den Meßprozeß,” Ann. Phys. 443, 322328 (1950).
72.S. Doplicher, “The measurement process in local quantum theory and the EPR paradox,” e-print arXiv:0908.0480 [quant-ph] (2009).
73.D. Buchholz, “Product states for local algebras,” Commun. Math. Phys. 36, 287304 (1974).
74.H. Halvorson, “Remote preparation of arbitrary ensembles and quantum bit commitment,” J. Math. Phys. 45, 4920 (2004).
75.M. Ozawa, “Canonical approximate quantum measurements,” J. Math. Phys. 34, 55965624 (1993).
76.D. Buchholz, S. Doplicher, and R. Longo, “On Noether’s theorem in quantum field theory,” Ann. Phys. 170, 117 (1986).
77.C. D’Antoni, S. Doplicher, K. Fredenhagen, and R. Longo, “Convergence of local charges and continuity properties of W-inclusions,” Commun. Math. Phys. 110, 325348 (1987).
78.C. C. Chang and H. J. Keisler, Model Theory, 3rd ed. (North-Holland, Amsterdam, 1990).
79.A. Robinson, Non-Standard Analysis (North-Holland, Amsterdam, 1966).
80. One may consider that we can prepare states only in bounded local regions, i.e., (normal) states on for some . This intuition is realized as the concept of local state.56 A local state T on with an inclusion pair of regions is defined as a unital CP map T on satisfying the following conditions, where such that : (i) T(AB) = T(A) B for all and ; (ii) there exists such that T(A) = φ(A)1 for all . On the other hand, we need states on to describe the given physical system. By using local states, states on can be regarded as local states with in the limit of tending to the whole space ℝ4. The authors believe that the use of local states would be helpful for developing measurement theory in local quantum physics in the future.

Data & Media loading...


Article metrics loading...



In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.


Full text loading...


Access Key

  • FFree Content
  • OAOpen Access Content
  • SSubscribed Content
  • TFree Trial Content
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd