Thursday, 5 January 2017

Informational and physical duality

John Campbell
January 2017

This post is an excerpt from the book: Darwin Does Physics.



 Many, if not all, complex structures found in nature have both physical and informational components. As we have seen biology, the brain and behaviour, as well as cultural processes, including science, may all be viewed in terms of a physical form and an internal informational model. While it has been understood for some time that information must have a physical form (1) this examination suggests that the converse is also true: physical form must have an informational description.

I have suggested that neither component may be considered more fundamental but rather should be considered as different aspects of inferential systems. Perhaps the best understood inferential system operating outside of the human sphere is biology. Here we have a deep understanding of the relationship between the informational and physical aspects of the system, the genotype and the phenotype. Neither may be considered as more fundamental; the genotype contains the ‘construction blue print’ for the phenotype but it is the generational experiences of the phenotype which is responsible for the accumulation of knowledge in the genotype.

It appears that nature has used this same mechanism of knowledge accumulation throughout numerous areas of scientific subject matter. If this is the case then we must accept that this unifying concept is more fundamental than the many divergent details studied within the various subject matters.

It is perhaps suggestive that this same duality has been proposed at the foundations of physical theory: the holographic principle. The holographic principle claims that the physics which takes place within a volume of space-time is completely described by information residing on the boundary of that space-time. The physical form of the information on the boundary has not been discovered nor have many of the details of the relationship between this information and the physics in the volume. In attempting to reduce our ignorance in this area we might consider that the many general principles found to operate in other natural inferential or Darwinian systems may serve as a guide to this newly discovered informational/physical duality within physics.

I will make this radical leap and consider the possibility that the analogy between biological and quantum systems may be rather exact. To aid the development of this view I will name the physical form of the quantum wave function as the systems quenome which is composed of quenes (pronounced ‘queens’). This nomenclature is in the tradition of genes, memes and temes suggested by Dawkins and Blackmore (2; 3; 4). I also propose the name ‘quantum phenotype’ for the physical structures of a quantum system. Quantum phenotypes would include objects such as quarks, atoms and molecules.

More generally, following Zurek, I will consider the transformation of the quenome into its phenotype as taking place through the process of decoherence. Thus classical reality may be considered the quantum phenotype.

An immediate insight offered by this analogy is that the Turing-Church-Deutsch principle (5), which states that any quantum system may be simulated to any degree of accuracy by a program written in qubits or quenes, is no more mysterious than the statement that an organism may be simulated to any degree of accuracy by a program written in genes. In both cases the conclusion follow naturally from the understanding that both phenotypes are constructed from their underlying ‘programs’.

Very recent research describes holographic information in terms of ‘entanglement entropy’. This concept combines the physical property of entanglement with the informational property of entropy. Quantum systems having a number of component subsystems may be entangled in the sense that a single wave function describes the complete system. While this description may be complete and unambiguous it fails to completely describe the subsystems. The amount of ignorance of the wave function concerning the subsystems is named entanglement entropy. As Zurek describes it (6):

This is a signature of entanglement that allows the state to be known ‘as the whole’, while states of subsystems are unknown.

The quantum wave function or quenome predicts properties of quantum systems from within the vast space of possibilities making up Hilbert space. This space is especially vast for many bodied entangled systems. For instance the dimension of the Hilbert space describing a litre of gas has 10230 dimensions. This unimaginably large number is 10150  times larger than the number of atoms in the universe. Obviously this number of possibilities must be drastically reduced before it can be of any predictive use.

We may conceive of biology in a similar manner. The human genome is coded in about 3 billion (3,000,000,000) base pairs, each of which may be 1 of 4 possible molecules. Thus the number of possibilities is 43,000,000,000 which is unimaginably bigger than even the quantum state. Progress may be made in making sense of this huge space by excluding the vast majority of these ‘construction blue prints’ which code for forms incapable of life. As life is a low entropy state we could make progress in selecting those genomes which code for viable humans by selecting only those which code for a low entropy state.  Futher reductions in our predictive space could be made if we were able to understand which quenomes were ‘fittest’ for the environment they would encounter.

Research using the mathematical tool of tensor networks proposes an analogous method of reducing the number of possibilities of the quenome. Tensor networks have been found to accurately describe many properties of entanglement entropy. It may appear obvious that the amount of information required to describe physics taking place in a volume should be proportional to that volume. However analysis using tensor networks and the basic physical principle of locality show that there is an extremely small subset of quenomes where the information required is proportional only to a surface area surrounding the volume (7):

And locality of interactions turns out to have important consequences. In particular, one can prove that low-energy eigenstates of gapped Hamiltonians with local interactions obey the so-called area-law for the entanglement entropy.




Figure 1:  The manifold of quantum states in the Hilbert space that obeys the area law scaling for the entanglement entropy corresponds to a tiny corner in the overall huge space.

As a surface area is almost infinitely smaller than the volume it encloses, the special group of quenomes described by the area law are analogous to viable genomes in that they describe low entropy systems.



Figure 2: The entanglement entropy (S) of a small subset of systems obeying the area law have the property that their entanglement entropy scales as does the area.

Thus it appears that a relatively tiny number of quenomes are distinguished through coding for systems having exceptionally low entropy; entropy which scales with area rather than with volume. As in biology these are the informational structures which code for phenotypes capable of an existence.

These advances within tensor network theory provide support for the basic informational/physical duality at the heart of the holographic principle (7):

Such a dimensional reduction strategy is nothing but an implementation, in terms of TN calculations, of the ideas of the holographic principle.

Systems which take the form of an informational/physical duality and are in a low entropy state beg the question of how this state is maintained in view of the second law of thermodynamics. The most straight forward answer is that the system must impose constraints on the increase of its entropy. As highlighted in our examination of many Darwinian systems the evolutionary vehicle may be understood as largely composed of a collection of constraints which have been discovered over evolutionary time and coded into the informational part or replicator of the system.

For example biological phenotypes may be considered as collections of adaptations which act as constraints against the high entropy of death. In turn, the information required for the construction of these adaptations is coded in the organism’s DNA. Biological science provides extensive details of the relationship between the informational and physical structures which compose an organism; the details of this relationship within quantum systems is only now beginning to be unravelled.

While the results of this research implies that we should consider viable quantum phenotypes as a special class of low entropy possibilities, we are left in the dark concerning the mechanisms which maintain these low entropy states in the face of the second law. Further we remain ignorant as to the physical form of entanglement entropy composing the quenome and the relationship between the quenomes and the quantum phenotypes.

Some progress has been reported in understanding how the physics of gravity emerges from its informational dual. As gravity is a force in the physical volume it imposes order and reduces entropy. However the informational dual to the physics contains only quantum descriptions but no description of gravity.

Using the tensor network approach it has been found that constraints on the entanglement entropy in the informational theory produce the gravitational equations of motion in the physical theory (8). Perhaps this is an example where information in the quenome is translated into constraints against entropy increase in the physics of the quantum phenotype.

This interpretation is encouraged by the fact that tensor networks describing the quenome in terms of entanglement entropy may be broken into components which code for different properties of the phenotype. The quenome may be considered as a collection of quenes. This biological analogy has not escaped the researchers; they explicitly consider these tensor components as acting within quantum systems very much as DNA does within biological systems (9):

we can think of this approach as decomposing the quantum state in ‘fundamental DNA blocks’, namely, tensors with less parameters. Reversely, the small tensors are the DNA of the wave function, in the sense that all the properties of the many-body quantum state can be read from the individual small tensors alone.



Figure 3:  Tensor network theory is able to decompose the quantum wave function Ψ into numerous simple components of tensor networks which researchers compare to DNA.

Perhaps surprisingly, more accurate predictions are possible on the makeup of the next generation in quantum theory than in biology. This is because the quenome, as described by the quantum state vector, evolves according to the Schrödinger equation which is a function of the systems Hamiltonian. The Hamiltonian is a mathematical description of the interaction between the system’s self-energy and sources of potential energy in the environment that are capable of influencing the system. This description is succinct because there are only four forces of nature which can serve as sources of potential energy in the system’s environment.

Schrödinger’s equation serves to keep the system’s quenome synced with its environment in such a manner that the informational content of the quenome is capable of predicting the most likely information which can be transferred from the system to its environment; information capable of surviving in that environment.

Unfortunately, because the relationship between genomes and their environments is much more complex there is no simple equivalent to the Schrödinger equation in biological theory and predictions concerning the makeup of the next generation are less accurate. However although quantum theory accurately predicts which quantum states can survive in the classical environment its explanation does not extend beyond mathematics; the underlying reality remains obscure.

Within biology we have good models for explaining those phenotypes which can survive in a given environment. We understand much of operation of the vast chemical and behavioural adaptations which the biological phenotype employs to make a living in its environment. Most starkly we understand many details of the spike in entropy that accompanies the death of an organism.

This level of detailed understanding is not available in quantum theory. However Zurek’s principle of the ‘predictability sieve’ states that a common characteristic of the quantum phenotypes which survive is that they are in the lowest entropy states available. The high entropy forms are ones that do not survive.

Perhaps research investigating the methods used by surviving quantum phenotypes to constrain their entropy production may provide a deeper explanation of quantum survival.

Natural selection provides an excellent model for the accumulation of knowledge in the genotype over evolutionary time as well as the transformation of this accumulated knowledge into the biological phenotype. In each generation the large number of tentative genetic forms is collapsed to those amongst them which can survive. We understand the survival plan coded in most of the possible genetic plans amounts to gibberish. Only those plans with the knowledge to constrain their entropy production and maintain their structures are realized in biological phenotypes which do survive.

The quantum situation may be similar. According to axiom 3, with each new piece of information copied to the environment through decoherence the quenome undergoes a radical jump to match that information. Quantum Darwinism interprets this collapse as parallel to the collapse of biological possibilities to those capable of survival.

Composite quantum phenotypes, such as atoms, are typically decohered by their environment almost continuously. Their ‘classical trajectory’, in terms of the information they pass to classical reality appears continuous and predictable as shown by the validity of Newtonian mechanics over a wide swath of experimental conditions. Thus we can understand quantum phenotypes as composing classical reality (10) and to be evolving in time and accumulating knowledge of their environments.

Such composite quantum systems may display entanglement, indeed it is probable that as the universe tends towards equilibrium entangled quantum systems are evolving in size and complexity. The single wave functions which describe these complex quantum systems contain the necessary knowledge for the existence of these systems. As it is posited that there was very little entanglement at the Big Bang it is plausible that this knowledge has accumulated over evolutionary time in a manner analogous to the evolution of biological knowledge.

This short survey of possible but speculative analogies between quantum and biological phenomena is only meant to be suggestive. However, as I have documented, the scientific literature of fundamental physics makes numerous analogies with biological processes. I will make one last argument for taking these analogies seriously.
It is widely understood that quantum theory may be used to provide descriptions of all entities in the universe, even the most complex ones. As noted by Zurek (11):

The consistent (or decoherent) histories framework is a mathematical formalism for applying quantum mechanics to completely closed systems, up to and including the whole universe.

Even biological organisms conform to quantum laws. Biological organisms may only employ those physical processes which are allowed by physics, chemistry and thermodynamics. They cannot use atoms with one proton and ten neutrons nor can they invoke processes which produce a net decrease of entropy; these are not allowed by the laws of physics.

The unique forms produced by biological processes are composed of building blocks provided by the laws of physics and are themselves physical structures which operate according to quantum principles. Biology is confined to those structures on which physical principals bestow survival. For instance even the structure of our fantastically complex DNA must be capable of physical survival prior to its potential to serve as a biological agent. At a physical level the quenome which models a DNA molecule must contain the knowledge necessary for its existence.

We might consider that chemistry was able to evolve to a level of complexity approaching the biological before this process came under the guidance of natural selection. The knowledge within the quenomes of these entities evolved through an exploration of possible complex chemical forms according to its own inherent mechanisms.

The fact that each of us is unique in the universe and may be uniquely identified by our DNA implies that the quantum wave function which describes us is also unique in the universe. Our biological evolution has been in parallel to the evolution of our wave function but at least in some sense the physical evolution is more fundamental.

Thus the knowledge in our DNA built up through evolutionary time via the processes of natural selection is mirrored by the knowledge contained in our quantum wave function. So here is least one case where quantum knowledge has evolved parallel to a well understood process of Darwinian knowledge evolution. The fantastically complex wave functions which describe biological organisms has evolved to its current state in parallel with the process of natural selection. It is likely that the quantum wave function describing biological phenomena is not unique but evolves in a typical fashion to other quantum phenomena. If quantum evolution is a consistent process over all phenomena then it is consistent with a Darwinian process.

We have a deep understanding of the process by which knowledge has been accumulated in our DNA but very little idea of how it accumulates in wave functions. In the face of our ignorance it seems plausible that an efficient path to understanding the evolution of quantum knowledge may be to take the biological analogy seriously and to consider quantum evolution as a Darwinian process.

Bibliography

1. Decoherence, einselection and the existential interpretation (the rough guide). Zurek, Wojciech H. s.l. : http://arxiv.org/abs/quant-ph/9805065, 1998, Philosophic Transactions of the Royal Society; vol. 356 no. 1743, pp. 1793-1821.

2. Dawkins, Richard. The Selfish Gene. s.l. : Oxford University Press, 1976.

3. Blackmore, Susan. The Meme Machine. Oxford, UK : Oxford University Press, 1999.

4. —. Genes, memes and temes. Susan Blackmore. [Online] [Cited: May 4, 2014.] http://www.susanblackmore.co.uk/memetics/temes.htm.

5. Quantum theory, the Church-Turing principle and the universal quantum computer. Deutsch, David. 1985, Proceedings of the Royal Society of London A 400 , 97-117., pp. 97-117.

6. Quantum Darwinism. Zurek, Wojciech H. s.l. : http://www.nature.com/nphys/journal/v5/n3/abs/nphys1202.html, 2009, Nature Physics, vol. 5, pp. 181-188.

7. A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States. Orus, Roman. s.l. : Annals of Physics 349 (2014) 117-158, http://arxiv.org/abs/1306.2164, 2014.

8. Gravitation from entanglement in holographic CFTs. Faulkner, Thomas, et al. s.l. : http://arxiv.org/abs/1312.7856.

9. Advances on Tensor Network Theory: Symmetries, Fermions, Entanglement and Holography. Orus, Roman. s.l. : Arxiv preprint.

10. Decoherence and the Transition form Quantum to Classical - Revisited. Zurek, Wojciech H. s.l. : http://arxiv.org/ftp/quant-ph/papers/0306/0306072.pdf, 2003.

11. The objective past of a quantum universe - Part 1: Redundant records of consistent histories. Riedel, C. Jess, Zurek, Wojciech H. and Zwolak, Michael. s.l. : arXiv preprint.

12. Area laws for entanglement entropy - a review. Eisert, J., Cramer, M. and Plenio, M. B.