Bounds on decoherence and error

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 57, Issue 2, 1998, Pages 840-844

  Schulman, L S ^a  

^a CLARKSON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041866738     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.57.840     Document Type: Article

References (23)

1. Among many references, see, for example, I. L. Chuang, R. Laflamme, P. W. Shor, and W. H. Zurek, Science 270, 1633 (1995). Note that for quantum computing purposes the entanglement discussed in the present article is relevant only to the extent that different states in the Hilbert space associated with the computation couple differently to the translational degrees of freedom.SCIEAS
2. L. S. Schulman, Time’s Arrows and Quantum Measurement (Cambridge University Press, Cambridge, 1997).
3. L. S. Schulman, Phys. Lett. A 211, 75 (1996). Note a misprint: “ε” there should be (Formula presented) (not (Formula presented)).PYLAAG
4. A. Shimony, Ann. N.Y. Acad. Sci. 755, 675 (1995).ANYAA9
5. By “decoherence” I mean loss of the primary wave function through entanglement with other degrees of freedom, hence the inability to interfere with portions of the wave function not so entangled. By “error” I mean a nonentangled wave function whose value is changed from that associated with idealized potential scattering.
6. For electromagnetic or other confinement this picture will need extension. However, the primitive underpinning of the derivation, momentum conservation, suggests that such an extension is possible.
7. A treatment neglecting the dynamical nature of the wall would generally omit the function Γ.
8. The validity of this assertion depends on the separation of the incoming and outgoing wave packets. To derive a quantitative measure of this I must be more explicit about the propagator in Eq. (1). Because of the hard wall, the coordinate (Formula presented) (Formula presented) is only defined for negative values (since I take the small particle to be coming from the left). It is more convenient, however, to extend the space to the entire real line and consider the initial wave packet to have consisted of two pieces, one coming from its actual source and one coming from the mirror image. (This is the method of images applied to the path integral 9.) To study the separation of the wave packets I ignore the small effect of the entanglement with the wall [this has been checked with Gaussian wave packets and only changes the outcome by (Formula presented)]. I thus ascertain whether with the naive calculation (treating the wall as a fixed potential) the incident and reflected wave packets separate. First, if the initial position is (Formula presented) then it must be the case that (Formula presented) where σ is the spread of the (Formula presented)-wave function. All that is left to check is that wave-packet spreading during the time it takes for the packets to separate does not overwhelm the effect of the relative velocity of the separating (incident and reflected, or source and image) wave packets. To see this, recall that for a free particle the time-(Formula presented) evolute of a particle with initial wave function (Formula presented) is (Formula presented) with (Formula presented) and (Formula presented) Small wave-packet spread implies (Formula presented) while having the incoming and outgoing packets separate from one another requires (Formula presented) Combining these gives the requirement (Formula presented) in order for our calculational method to be valid.
9. A. Auerbach and L. S. Schulman, J. Phys. A 30, 5993 (1997).JPHAC5
10. Allowing the integration to range over the whole line is another correlate of the method of images and the assumption that the initial wave packet was negligibly small at the wall. To show this I drop the complication of treating the wall dynamically. What one should calculate is (Formula presented) Since (Formula presented) vanishes for (Formula presented) (so far, I’ve not set (Formula presented) to zero), the integral over (Formula presented) can be run over the entire line. For the (Formula presented) integral we change (Formula presented) to (Formula presented) so that it now looks like a source at (Formula presented) At this point I set (Formula presented) and make use of the convenient wave function form given in Eq. (3).
11. Recall from Ref. 8 that (Formula presented) should be larger than one for our calculational method to apply.
12. M. M. Yanase, Phys. Rev. 123, 666 (1961).PHRVAO
13. Following Ref. 14 (and in agreement, up to overall constants, with Ref. 4), one can define the measure of entanglement either as (Formula presented) or as (Formula presented) with (Formula presented) in the second version. One then invokes the following mathematical result. Let (Formula presented) be an arbitrary (Formula presented) matrix and let (Formula presented) Then if (Formula presented) and (Formula presented) minimize L, they satisfy (Formula presented) and (Formula presented) From this it follows that (Formula presented) (Formula presented) and (Formula presented) the maximum eigenvalue of (Formula presented) Defining (Formula presented) (Formula presented) and (Formula presented) for a matrix (Formula presented) we can write (Formula presented)(Formula presented) It is also clear that (Formula presented) which is real, and finally (Formula presented) The equivalence of the definitions (Formula presented) and (Formula presented) can be seen by noting the correspondence of L and (Formula presented) with (Formula presented) playing the role of (Formula presented) I need to show that functions (Formula presented) and (Formula presented) that minimize (Formula presented) also maximize (Formula presented). For arbitrary, normalized (Formula presented) and (Formula presented) let (Formula presented) where γ is an arbitrary complex constant. If we adjust (Formula presented) to make (Formula presented) real, then (Formula presented) Taking γ to be real obviously can only reduce (Formula presented) Since the foregoing equation holds for any real γ, it is seen that maximizing (Formula presented) is the same as minimizing (Formula presented) It then follows that (Formula presented) etc.
14. L. S. Schulman and D. Mozyrsky (unpublished).
15. L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981).
16. For (Formula presented) one can also get (Formula presented) i.e., no entanglement, but this is not the physical situation emphasized in this article.
17. Disentanglement with spread matching could have been noted directly from (Formula presented) and does not require Ref. 4. However, the measure of the amplitude defect without matching does require those results.
18. There is of course Feynman’s variation on the two slit experiment [in R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, MA, 1965)], which relates uncertainties in a macroscopic object to putative measurements of a microscopic one. The corresponding restriction here would be that (Formula presented) (Formula presented) not be so small that the associated (Formula presented) destroy the small-system interference patterns that we seek. Our optimal Σ is far from such values. We estimate this kinematic effect as follows. Momentum uncertainty (Formula presented) in the big system means uncertainty (Formula presented) in the (velocity) transformation going into the center-of-mass frame. For the small system this velocity uncertainty gives a momentum uncertainty (Formula presented) [the prime on (Formula presented) distinguishes it from the momentum uncertainty in the original wave function, namely, (Formula presented)]. Taking (Formula presented) we find (Formula presented) Using (Formula presented) yields (Formula presented)
19. There have been suggestions that nature ought to evolve into coherent states. See W. H. Zurek, S. Habib, and J. P. Paz, Phys. Rev. Lett. 70, 1187 (1993).PRLTAO
20. The inverse mass relation was also found in 19. The energy demanded by dimensional analysis is there an oscillator frequency. It is not clear (to me) whether their (Formula presented)(Formula presented) and my Eq. (9) are related.
21. H. Grabert, P. Schramm, and G. Ingold, Phys. Rep. 168, 115 (1988). See in particular Table 2, p. 159.PRPLCM
22. The condition (Formula presented) in Ref. 8 was not needed for the disentanglement, only for the justification of wave-packet separation and as a limitation on spreading, so as to allow this separation.
23. L. S. Schulman, Ann. Phys. (N.Y.) 183, 320 (1988). APNYA6