In a note you wrote to me you mentioned that you believed that Peres had proved that nonlinear QM violated causality and allowed for superluminal signaling. N. Gisin also thought he had demonstrated inconsistency between nonlinear QM and special relativity [1].
Nonlinear QM does not automatically conflict with causality, though. Several versions of deterministic nonlinear QM can peacefully coexist with special relativity. This has been demonstrated by Joseph Polchinski [2], Thomas Jordan [3] and Marek Czachor [4,5] for selected classes of nonlinear and nonlocal SE's which preserve the apparent locality of dynamics of subsystems. In these versions of nonlinear QM, no local measurement on a compound system affects the evolution of separated parts of the system in a way that transmits information.
Semiclassical versions of gravity and electrodynamics in which matter is quantized but in which either the gravitational and/or electromagnetic field is classical also represent yet other versions of nonlinear QM. Branches of the wave function interact with each other via a classical field which does not propagate disturbances superluminally. Consequently, these theories are inherently consistent with special relativity as shown by L. Bonilla and F. Guinea [6,7] for semi-classical gravity. There are many papers arguing that semiclassical electrodynamics can reproduce the successful experimental predictions of QED. For example, see [8-10].
Yet another approach to nonlinear QM which I think can made consistent with causality is based on quantum friction. I'll discuss this again in my next e-mail letter.
The interesting thing about nonlinear QM theories is that they would permit the (probably metaphorical) Everett worlds to interact, or collaborate to produce a final "collapsed" state. This might allow a quantum computer to harness an enormous amount of parallelism (which a classical computer of the same mass could not access) by exploring different options in different branches of the wave function. A quantum computer evolving according to the standard linear SE can only combine these results in the final reduction step (ie - at measurement). This severely limits its effective throughput and with a few exceptions (like factoring integers) linear quantum computers cannot (on average) outperform classical ones.
Even more interesting is a phenomenon pointed out by Polchinski [2] and by Czachor [4]. In the versions of nonlinear QM they considered, an observer can measure a quantum system so as to statistically bias the "collapse" of its wave function. I have not been able to contrive a specific scheme yet, but I believe that it would permit a scenario like this:
An observer starts a quantum computer in a superposition of states in which it is computing a function, F, for arguments 1,2,3, ..., N in N separate (metaphorical) Everett worlds.
The quantum computer evolves without observation for a length of time sufficient for its classical analog to have computed the value F(i) for any of the values i = 1,2,3, ..., N (i.e. worst case timing)
The observer decides after the fact that he wants the value F(i). Via an appropriate measurement he collapses the quantum computer's wave function on F(i) eliminating all the other branches.
This would amount starting a computer on Monday and then coming back to it on Friday and telling it what it "has done" (as long as the task selected would not take longer than 5 days to run on its classical counterpart) as opposed starting it running on a well-defined known task. The stipulation that the quantum computer's evolution not be perturbed evades the famous "grandfather paradox" of time travel.
Probably too good to be true.
[1] "Stochastic quantum dynamics and relativity"
- N. Gisin Helvetica Physica Acta, vol. 62, (1989), pp. 363-371
[2] "Weinberg's Nonlinear Quantum Mechanics and the Einstein- Podolsky-Rosen Paradox"
- Joseph Polchinski Physical Review Letters, vol. 66, (28 Jan 1991), pp. 397-400
[3] "Reconstructing a Nonlinear Dynamical Framework for Testing Quantum Mechanics"
- Thomas F. Jordan Annals of Physics, vol. 225, (1993), pp. 83-113
[4] "Elements of Nonlinear Quantum Mechanics (I): Nonlinear Schroedinger equation and two-level atoms"
- Marek Czachor HEP preprint: QUANT-PH-9501007
[5] "Elements of Nonlinear Quantum Mechanics (II): Triple bracket generalization of quantum mechanics"
- Marek Czachor HEP preprint: QUANT-PH-9406174
[6] "Reduction of the wavepacket through classical variables"
- L.L. Bonilla and F. Guinea Physics Letters B271, (1991), pp. 196-200
[7] "Collapse of the wave packet and chaos in a model with classical and quantum degrees of freedom"
- L.L. Bonilla and F. Guinea Physical Review A, vol. 45, (1 June 1992), pp. 7718-7728
[8] "The Quantum Theory of Self-organizing Electrically Charged Matter: Solutions of the Fundamental Equation of Motion"
- V.P. Oleinik CERN preprint: P00022650 (1993)
[9] "An Approach to Finite Non-Perturrbative Quantum-Electrodynamics"
- Asim O. Barut in "Proc. 2nd Int'l. Symp. on the Foundations of Quantum Mechanics: Tokyo, 1986" pp. 323-327
[10] "Understanding the Self-Interaction of Charged Particles"
- A.J. DeGregoria Nuovo Cimento B53, (1979), pp. 117-133
Regards,
Lou Pagnucco (lpagnucco@delphi.com)