John O. Campbell

January 2017

# 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 10

^{230 }dimensions. This unimaginably large number is 10^{150 }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 4

^{3,000,000,000}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 Î¨ 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 SchrÃ¶dinger 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.

SchrÃ¶dinger’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
SchrÃ¶dinger 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.**