McFadden and Al-Khalili's reply: quant-ph/0110083.
McFadden and Al-Khalili's reply does nothing to alter my opinion. They introduce a model of particles in a double potential well, but this is merely another analogy. The issue is about the plausibility of quantum coherent mechanisms serving specific purposes in biological systems, given our present understanding of quantum mechanics and of the known interactions between particles in such systems.
The most interesting point raised by McFadden and Al-Khalili's reply comes in the paragraph in which, writing about degrees of freedom in a living cell which they claim could maintain quantum coherence, they say,
The degree of isolation of those degrees of freedom will depend on the leakage of information — from the cell to its environment — that would betray the positions of the particles involved.
The problem here is the common misconception that decoherence can only occur when information leaks to the environment in a usable or measurable form. This misconception arises through familiarity with simple systems like the two slit experiment in which interference is destroyed only when there is information about which slit was traversed. Similarly, in the Schroedinger cat experiment, the radioactive atom passes information to the cat. More generally, if we have a globally pure state split between two subsystems, the Schmidt decomposition can be used to show that the entropies of the two subsystems are always equal, and we can interpret these entropies as being equal information lost and gained. Nevertheless, in a large system, small amounts of entropy are easily hidden, because, in a large system, orthogonality between wavefunctions is easily achieved by many small differences spread over the system.
An environment as large as a cell or its surroundings can therefore reasonably be modelled as a heat bath. In that case, information can leak in the form of heat and this will betray nothing. Indeed, on page 44 of D. Giulini, E. Joos, C. Kiefer, J. Kupsch, I.-O. Stamatescu, and H.D. Zeh, “Decoherence and the Appearance of a Classical World in Quantum Theory” (Springer, 1996) — Joos gives a simple and entirely explicit model in which the state of a system in contact with a heat bath becomes decoherent under a global unitary map without the state of the environment changing at all. Correlation between system and environment is still the driving force of the decoherence, but the effect of the correlation is not discernable from the environment alone.
In my review I argue that it is an error for McFadden and Al-Khalili to estimate the timescales on which coherence can persist for many-particle spatial wavefunctions in biological systems by using spin-lattice relaxation times measured by NMR. I am surprised that they try to defend this point. What is measured in NMR is the relaxation of nuclear spin states. NMR is useful in biological systems precisely because, in a fluid, nuclear spins have extremely weak interactions with their environment compared with nuclear positions. A proton spinning in a given direction can maintain coherence in spin direction far longer than it can maintain coherence in position. Protons (even “hydrogen ions” which are actually strongly hydrated) are wrapped in electron clouds which protect the nuclear spin. The position of the proton however is buffeted by all its neighbours' modes of vibration and translation. Timescales of tens of seconds are indeed possible for spin relaxation, but I would expect coherence in position in living systems, even for individual particles, typically to relax on the vibrational mode timescales of 10-13 seconds.
McFadden and Al-Khalili refer to a paper by Horsewill, Jones, and Caciuffo (Science 291, pp 100-103, (2001)), in which NMR relaxation times were indeed used to measure the timescales of coherent proton tunnelling. However, not only was this work on a solid system below 80K, but the proton tunnelling frequency of 35 MHz (corresponding to a timescale of 10-8 seconds) was measured by looking at how the NMR relaxation timescale (between 8 and 50 seconds) changed as a function of magnetic field, with the peak magnetic field interpreted as corresponding to its Lamor frequency.
In the case of the protons considered by McFadden and Al-Khalili, on the timescales in which they are interested, position is correlated, through DNA transcription, with biosynthetic activity across the cell. They are therefore actually dealing with the entire multi-particle state of a macroscopic portion of warm fluid. The proton position is analogous to the decay state of the radioactive atom in the Schroedinger cat experiment — as soon as correlation to the state of the cat has occured, the relevant system for consideration of coherence has to include the cat. By the time the different transcriptions have reached the point of placing different amino acids at the end of a peptide chain, the differences in molecular species will have affected the positions of all the particles which those amino acids are in contact with and all the particles which the different RNA molecules are in contact with and the effects will ripple out at the speed of sound to the entire cell and beyond. No boy dances exactly the same dance with two different partners, and if he leads the brunette on to the crowded floor rather than the blonde, we wouldn't expect anyone in the room to be in exactly the same place at the end of the dance.
While it is not impossible that there might be specific multi-particle degrees of freedom of biological relevance within a cell which maintain coherence over long periods, such degrees of freedom would have to be isolated, for those periods, from correlation to other degrees of freedom. This requires, in particular, isolation from random thermal influences both internal and external to the cell and certainly requires isolation from biochemical processes. None of the examples of observed coherent behaviour referred to by McFadden and Al-Khalili give me any reason to change my opinion of their proposals, because these examples concern systems which are very small scale, or at low temperature, or very precisely configured, or spatially uniform. None of these descriptions is appropriate for protein transcription.
McFadden and Al-Khalili say “our decoherence rate estimate was for only a single proton whereas a mutation will involve many more particles”. It is precisely because decoherence rates tend to increase so fast with the number of particles that quantum computers involving more than a few qubits are so hard to build.