Skip to main content
SpringerLink
Log in

The Unruh-DeWitt Detector and Entanglement Harvesting

  • Chapter
  • First Online:
Detectors, Reference Frames, and Time

Part of the book series: Springer Theses ((Springer Theses))

Abstract

As a model of a two-level quantum system interacting with a quantum field, the Unruh-DeWitt detector is introduced and physically motivated. Both the transition probability and transition rate of this detector model are defined and expressed to leading order in terms of the vacuum Wightman function for a scalar field living on a curved spacetime. The entanglement harvesting protocol is introduced in which two detectors begin in a separable state and as a result of their local interaction with the field become entangled. This entanglement is interpreted as being extracted from entanglement present in the vacuum state of the field and is quantified by various measures. The measurement model induced by a collection of such detectors is constructed, identifying the POVMs associated with the field that such a collection of detectors measures.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In the interaction picture, the interaction Hamiltonian is \(H_{I} := e^{iH_0t} H_{int}e^{-iH_0 t}\).

  2. 2.

    The diagonal elements of H I vanish because \( \left < 0 | \mathbf {X} | 0 \right > = \left < 1 | \mathbf {X} | 1 \right > =0\), which comes from the fact that ψ 0(x) and ψ 1(x) are symmetric around x = 0 due to the Coulomb interaction between the nucleus and electron of the atom being symmetric around x = 0.

  3. 3.

    As stated in [1], a local realist theory is one where physical properties are defined prior to and independent of measurement, and no physical influence can propagate faster than the speed of light.

  4. 4.

    The Svetlichny inequality is a Bell-like inequality whose violation is sufficient but not necessary for genuine tripartite nonlocal correlations [47].

  5. 5.

    The partial transpose of the state \(\rho _{AB} \in \mathcal {S} \left (\mathcal {H}_A \otimes \mathcal {H}_B\right )\) with respect to \(\mathcal {H}_A\) is \(\rho _{AB}^{\Gamma _A}:=\left [{T} \otimes \mathcal {I}\right ] \left ( \rho _{AB} \right )\), where \({T}:~\mathcal {S}(\mathcal {H}_A)~\to ~\mathcal {S}(\mathcal {H}_A)\) is the transposition map.

  6. 6.

    An observable is defined on a measurable space \((\Omega ,\mathcal {F})\), where Ω is a sample space and \(\mathcal {F}\) is a collection of subsets of Ω (\(\mathcal {F}\) is a σ-algebra); \((\Omega ,\mathcal {F})\) is the outcome space of the observable. An observable A is a map \(A: \mathcal {F} \to \mathcal {E}\left (\mathcal {H}\right )\), where \(\mathcal {F}\) is the space of possible outcomes of a measurement of the observable A on a system associated with the Hilbert space \(\mathcal {H}\), such that A is a positive operator valued measure (POVM). A POVM is a map \(A: \mathcal {F} \to \mathcal {E}\left (\mathcal {H}\right )\) such that (a) A(∅) = 0, (b) A( Ω) = I, and (c) A(∪i X i) =∑i A(X i) for any sequence {X i} of disjoint sets in \(\mathcal {F}\); for \(X \in \mathcal {F}\), A(X) are referred to as POVM elements. In other words, a mapping \(A: \mathcal {F} \to \mathcal {E}\left (\mathcal {H}\right )\) is a POVM if and only if the mapping \(X \mapsto \operatorname {\mathrm {tr}} \left [ \rho A(X)\right ]\) defines a probability measure for every state \(\rho \in \mathcal {S}\left (\mathcal {H}\right )\).

  7. 7.

    From [20]: A self-adjoint operator \(W \in \mathcal {L}_s \left (\mathcal {H}_A\otimes \mathcal {H}_B\right )\) is an entanglement witness if W is not a positive operator but \(\left < \psi \right |\left < \phi \right | W \left | \psi \right >\left | \phi \right > \geq 0\) for all factorized vectors \(\left | \psi \right >\left | \phi \right > \in \mathcal {H}_A\otimes \mathcal {H}_B\). We say that an entangled state ρ is detected by W if \( \operatorname {\mathrm {tr}}[\rho W] < 0\).

References

  1. J. Ȧke Larsson, Loopholes in Bell inequality tests of local realism. J. Phys. A 47, 424003 (2014)

    Google Scholar 

  2. A.M. Alhambra, A. Kempf, E. Martín-Martínez, Casimir forces on atoms in optical cavities. Phys. Rev. A 89, 033835 (2014)

    Article  ADS  Google Scholar 

  3. M. Ali, A.R.P. Rau, G. Alber, Quantum discord for two-qubit X states. Phys. Rev. A 81, 042105 (2010)

    Article  ADS  Google Scholar 

  4. L. Amico, R. Fazio, A. Osterloh, V. Vedral, Entanglement in many-body systems. Rev. Mod. Phys. 80, 517 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  5. D. Beckman, D. Gottesman, A. Kitaev, J. Preskill, Measurability of Wilson loop operators. Phys. Rev. D 65, 965022 (2002)

    Article  MathSciNet  Google Scholar 

  6. J.S. Bell, On the Einstein Podolsky and Rosen paradox. Physics 1, 195 (1964)

    Article  MathSciNet  Google Scholar 

  7. J.S. Bell, On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447 (1966)

    Article  ADS  MathSciNet  Google Scholar 

  8. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  9. N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982)

    Book  Google Scholar 

  10. L. Bombelli, R.K. Koul, J. Lee, R.D. Sorkin, Quantum source of entropy for black holes. Phys. Rev. D 80, 373 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  11. D. Bruß, G. Leuchs, Lectures on Quantum Information (Wiley-VCH, Weinheim, 2007)

    MATH  Google Scholar 

  12. C. Callan, F. Wilczek, On geometric entropy. Phys. Lett. B 33, 55 (1994)

    Article  ADS  MathSciNet  Google Scholar 

  13. H. Casini, M. Huerta, A finite entanglement entropy and the c-theorem. Phys. Lett. B 600, 142 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  14. H. Casini, M. Huerta, Entanglement entropy in free quantum field theory. J. Phys. A 41, 504007 (2009)

    Article  MathSciNet  Google Scholar 

  15. B.S. DeWitt, Quantum gravity: the new synthesis, in General Relativity: An Einstein Centenary Survey (Cambridge University Press, Cambridge, 1979), pp. 680–745

    Google Scholar 

  16. F. Dowker, Useless qubits in “relativistic quantum information” (2011). arXiv:quant-ph/1111.2308

    Google Scholar 

  17. A. Einstein, B. Podolsky, N. Rosen, Can quantum mechanical description of physical reality be considered complete. Phys. Rev. Lett. 47, 777 (1935)

    ADS  MATH  Google Scholar 

  18. C.J. Fewster, R. Verch, Algebraic quantum field theory in curved spacetimes, in Advances in Algebraic Quantum Field Theory (Springer, Cham, 2015), pp. 125–189

    MATH  Google Scholar 

  19. S.J. Freedman, J.F. Clauser, Experimental test of local hidden variable theories. Phys. Rev. Lett. 28, 938 (1972)

    Article  ADS  Google Scholar 

  20. T. Heinosaari, M. Ziman, The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement (Cambridge University Press, Cambridge, 2012)

    MATH  Google Scholar 

  21. S. Hollands, R.M. Wald, Axiomatic quantum field theory in curved spacetime. Commun. Math. Phys. 293, 85 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  22. M. Horodecki, P. Horodecki, R. Horodecki, Separability of mixed states: necessary and sufficient conditions. Phys. Lett. A 223, 1 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  23. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  24. J. Louko, A. Satz, Transition rate of the Unruh-DeWitt detector in curved spacetime. Classical Quantum Gravity 25, 055012 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  25. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995)

    Book  Google Scholar 

  26. E. Martín-Martínez, E.G. Brown, W. Donnelly, A. Kempf, Sustainable entanglement production from a quantum field. Phys. Rev. A 88, 052310 (2013)

    Article  ADS  Google Scholar 

  27. E. Martín-Martínez, M. Montero, M. del Rey, Wavepacket detection with the Unruh-DeWitt model. Phys. Rev. D 87, 064038 (2013)

    Article  ADS  Google Scholar 

  28. E. Martín-Martínez, A.R.H. Smith, D.R. Terno, Spacetime structure and vacuum entanglement. Phys. Rev. D 93, 044001 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  29. M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2010)

    Book  Google Scholar 

  30. A. Osterloh, L. Amico, G. Falci, R. Fazio, Scaling of entanglement close to a quantum phase transitions. Nature 416, 608 (2002)

    Article  ADS  Google Scholar 

  31. A. Peres, Separability criterion for density matrices. Phys. Rev. Lett. 77, 1413 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  32. A. Pozas-Kerstjens, E. Martín-Martínez, Harvesting correlations from the quantum vacuum. Phys. Rev. D 92, 064042 (2015)

    Article  ADS  Google Scholar 

  33. A. Pozas-Kerstjens, E. Martín-Martínez, Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms. Phys. Rev. D 94, 064074 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  34. B. Reznik, Entanglement from the vacuum. Found. Phys. 33, 167 (2003)

    Article  MathSciNet  Google Scholar 

  35. B. Reznik, A. Retzker, J. Silman, Violating Bell’s inequalities in vacuum. Phys. Rev. A 71, 042104 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  36. S. Ryu, T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT. Phys. Rev. Lett. 96, 181602 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  37. G. Salton, R.B. Mann, N.C. Menicucci, Acceleration-assisted entanglement harvesting and rangefinding. New J. Phys. 17, 035001 (2015)

    Article  ADS  Google Scholar 

  38. S. Schlicht, Considerations on the Unruh effect: causality and regularization. Class. Quant. Grav. 21, 4647 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  39. E. Schroödinger, Discussion of probability relations between separated systems. Math. Proc. Camb. Philos. Soc. 31, 555–563 (1935)

    Article  ADS  Google Scholar 

  40. M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997)

    Book  Google Scholar 

  41. J. Silman, B. Reznik, Three-region vacuum nonlocality (2005). arXiv:quant-ph/0501028

    Google Scholar 

  42. R.D. Sorkin, Impossible measurements on quantum fields, in Directions in General Relativity, ed. by B.L. Hu, T.A. Jacobson (Cambridge University Press, Cambridge, 1993), pp. 293–305

    Chapter  Google Scholar 

  43. M. Srednicki, Entropy and area. Phys. Rev. Lett. 71, 666 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  44. S.J. Summers, R. Werner, The vacuum violates Bell’s inequalities. Phys. Lett. A 110, 257 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  45. S.J. Summers, R. Werner, Bell’s inequalities and quantum field theory. I. General setting. J. Math. Phys. 28, 2440 (1987)

    ADS  MathSciNet  MATH  Google Scholar 

  46. S.J. Summers, R. Werner, Bell’s inequalities and quantum field theory. II. Bell’s inequalities are maximally violated in the vacuum. J. Math. Phys. 28, 2448 (1987)

    MathSciNet  MATH  Google Scholar 

  47. G. Svetlichny, Distinguishing three-body from two-body nonseparability by a Bell-type inequality. Phys. Rev. D 35, 3066 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  48. W.G. Unruh, Notes on blackhole evaporation. Phys. Rev. D 14, 870 (1976)

    Article  ADS  Google Scholar 

  49. A. Valentini, Non-local correlations in quantum electrodynamics. Phys. Lett. A 153, 321 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  50. G. Ver Steeg, N.C. Menicucci, Entangling power of an expanding universe. Phys. Rev. D 79, 044027 (2009)

    Article  ADS  Google Scholar 

  51. G. Vidal, R.F. Werner, A computable measure of entanglement. Phys. Rev. A 65, 032314 (2002)

    Article  ADS  Google Scholar 

  52. G. Vidal, J.I. Latorre, E. Rico, A. Kitaev, Entanglement in quantum critical phenomena. Phys. Rev. Lett. 90, 227902 (2003)

    Article  ADS  Google Scholar 

  53. C. Weedbrook, S. Pirandola, R. García-Patrón, N.J. Cerf, T.C. Ralph, J.H. Shapiro, S. Lloyd, Gaussian quantum information. Rev. Mod. Phys. 84, 621 (2012)

    Article  ADS  Google Scholar 

  54. W.K. Wootters, Entanglement of formation of an arbitrary state of two qubits. Phys. Rev. Lett. 80, 2245 (1998)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics and Astronomy, Dartmouth College, Hanover, NH, USA

    Alexander R. H. Smith

Authors
  1. Alexander R. H. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Smith, A.R.H. (2019). The Unruh-DeWitt Detector and Entanglement Harvesting. In: Detectors, Reference Frames, and Time. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-11000-0_3

Download citation

Publish with us

Policies and ethics

Search

Navigation