Quantum Life and Quantum Cognition

With the development of quantum computation, quantum game theory and decoherence we are confronted with some fundamental problems that are at the heart of quantum theory and biology, specifically, our views on the following two questions must be considered in a new light:

• What is life?
  
• What is consciousness?

This is the first part of a two-part paper being published at this blog. In this part we deal with the first question, and will address some of the consequences of quantum computation and decoherence, leading to the argument that the border between the living and the non-living must be revised, and that quantum computation implies that a quantum system can already be considered as a living, individuated structure. This implication may seem, at first sight, and in a superficial perspective, as surprising, but the argument is straightforward enough, being the consequence of a standard conception of what a living organism is and of the basic results from decoherence and quantum information theory.

Quantum life?

The question - what is life? - is far from trivial. A very rough rule of thumb for defining a living organism can be introduced in terms of the following criteria:

1. It must be an individuated structure;
2. It must possess a metabolism;
3. Its activity is directed towards its homeostasis, perpetuating of the life of the organism for a sustainable period of time.

The research developed in quantum computation has shown us that a quantum information processing system can give rise to individuated structures that become computing subsystems with stable degrees of freedom.

Furthermore, decoherence tells us that, given a sufficiently complex isolated quantum system (with a Hilbert space with a dimensionality greater than three), there are quantum logical gates that produce a separation of the system in different subsystems, that correspond to degrees of freedom that are measured by other subsystems. Each subsystem becomes completely described by the local degrees of freedom selected by the quantum logical gate, its local state becoming diagonal with respect to all local observables, so that the subsystem gains a border in its potential reality and acquires an actuality due to the measurement, an actuality which is described by a probabilistic actualization of the values for the local degrees of freedom that characterize the system.

Since relative decoherence, as proposed by Gonçalves and Golçalves (2007) holds in this case, it follows that classical expectations can be formed for all the local observables, so that the system is completely individuated in its physicality, both actual and potential.

Furthermore, it possesses a metabolism. The word metabolism comes from the Greek metaballein, which means change, or exchange. As Schrödinger argued in his book (What is life?) the metabolism of the living organism can be considered as an exchange of entropy, and, furthermore, given the relation between entropy and information, it can be considered as an exchange of quantities of information. The organism is maintaining its integrity through actively processing the environment and feeding upon negative entropy and exporting positive entropy.

Let us pick up this last idea, and, for purely expository purposes, consider a simple example of an isolated system of three qubits. We can subdivide this system into three subsystems: S0, S1 and S2. Let us assume that S0 is our system of interest and that initially we have the state:

Now, let us introduce the following state transition:

Since the state transition was unitary, information was conserved, which means that the von Neumann entropy (measured in qubits) is left unchanged at zero.

Now, we know that the joint von Neumann entropy of two quantum systems can be divided as follows:







The last term in the sum is the conditional entropy. As addressed by Horodecki et al. (2005), the conditional entropy of a quantum system A on a second quantum system B, can be measured as the amount of information that A had to send B, in order for B to have a complete information about A, that is, it measures the "amount of ignorance" that B has about A. For the present case we have:






Taking the variation, and knowing that the initial entropy of S0 and the initial conditional entropy are both zero, we obtain the result:

Now, since the state transition is unitary, it follows that the joint entropy is zero, which means that the increase in entropy on the system S0 is exactly matched by the information that the system gains on the environment. Indeed, because the conditional entropy is the amount of ignorance that the system has about its environment, the fact that it is negative and that it exactly matches the increase in entropy of the system, means that the entropy increase in the system is compensated by an increase in negative entropy that is associated with an increase in the information that the system has on its environment. We should stress that the exact same result would be obtained if we had considered S1+S2 as the system of interest and S0 as the environment!

This last statement should not surprise us, basically what is being stated is that if you consider the system Y to be the environment of a system X, then, we see the entropy of X increasing, and being compensated by a simultaneous increase in conditional entropy due to entanglement.

There is a clear exchange of information taking place here, due to the formation of correlations. This type of metabolism will be addressed again in the second part of the present work.

Taking, now, into account Kribs' work on noiseless subsystems (see bibliography for references), if there is a set of degrees of freedom, which leads to a set of subsystems, that are resilient to decoherence with respect to an environment of other subsystems, then, even if there are parts of this resilient set that become entangled with each other, on the whole set, the information is conserved. Considering this, we have a self-sustained information conserving system, in which the information inside the system may be locally exchanged, but, on the whole, it is conserved. The system's whole is self-sustained in its computation, and can be considered in its activity as maintaining its internal millieu in the form of local information exchanges. That is, the system's computing activity is simultaneously metabolic (in the form of the internal exchanges of information) and it conserves information, which means that it is a self-sustained computing system, that actively maintains a state of homeostasis.

Furthermore, even if we consider interactions between subsystems, as long as a subsystem remains individuated in its degrees of freedom, and interacts with other systems as such, then, we have a network of metabolic exchanges and computations in which the subystem, even if open, remains individuated as a system, during an indefinite amount of time (depending on the case).

This research on noiseless subsystems and the interactions between them is a key research problem for the Bilson-Thompson et al. (2006) proposal.

This completes the three conditions that were reviewed above, as making up a rule of thumb to define a living organism.

Bibliography

Beny, C.; Kempf, A.; Kribs, S. W., 2007, Quantum error correction of observables. arXiv:0705.1574

Bilson-Thompson, S.O.; Markopoulou, F.; Smolin, L., 2006, Quantum gravity and the standard model. hep-th/0603022

Gonçalves, C.P., and Gonçalves, C., 2007, An Evolutionary Quantum Game Model of Financial Market Dynamics - Theory and Evidence. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=982086, http://mpej.unige.ch/mp_arc/c/07/07-89.pdf

Horodecki, Michael; Oppenheim, Jonathan; Winter, Andreas, 2005, Quantum information can be negative. quant-ph/0505062

Kribs, D.W., and Spekkens, R.W., 2006, Quantum Error Correcting Subsystems are Unitarily Recoverable Subsystems. quant-ph/0608045

Kribs, D.W., and Markopoulou, F., 2005, Geometry from quantum particles. gr-qc/0510052