Information theory is one of the great mathematical breakthroughs of the 20th century. It has allowed to engineer the modern communication networks that we use and rely on almost every day. Since its inception, the electronic components used to store and transmit information have continuously shrunk in size, down to a length scale where quantum mechanical effects start to play a role. This has necessitated an extension of the classical information theory as pioneered by Shannon to quantum mechanical systems. This combination of theories is known as quantum information theory.
One of the main topics in quantum information theory is the calculation of channel capacities of quantum mechanical communication channels. The capacity of a communication channel is the amount of information that can reliably be sent through it per unit of time, the medium being a quantum mechanical particle in this case. In practical applications, it is an important problem to understand the influence of noise on this capacity. The simplest assumption on the type of noise at each transmission through the channel is that it is independent of past transmissions. This is the setting in which quantum channel capacities are most often calculated. This assumption however comes down to a decorrelation time of the noise that is much shorter than the time between subsequent transmissions. In high frequency communication channels the noise source can however retain a memory of past noise from transmission to transmission (Kretschmann & Werner, 2005).
In the work that I performed during my PhD under supervision of Mark Fannes, I have calculated capacities of a model channel that does feature noise correlations in time, the noise intensity being regulated by an underlying classical Markov process. We have shown how the capacity of this memory channel is related to the entropy density of a classical hidden Markov process. We have used numerical methods to efficiently calculate this entropy density and as a result determined the capacity of the quantum memory channel (Akhalwaya et al., 2009; Wouters et al., 2009). The numerics demonstrate an increase of the channel capacity with stronger noise correlations, a property that I have proven in my PhD thesis (Wouters, 2011). This result indicates applications in the design of communication channels and suggests that channels capacity can be improved by increasing the noise correlation.~This counterintuitive result can be understood by considering that correlated noise is in a sense more predictable and therefore easier to counteract.~
In another work we have extended the concept of a conditional state space from classical probability theory to the setting of quantum mechanics (Fannes & Wouters, 2009). The concept can be used to construct quantum analogues of classical hidden Markov processes by specifying that they should have a finite dimensional conditional state space, just like their classical counterparts. Some of these finitely correlated states are also ground states of certain quantum spin chains (Fannes et al., 1992). This new characerisation of their quantum correlations could prove useful in describing their physical properties. We have furthermore shown that the conditional space can be used to determine in a new geometrical way whether a quantum state is entangled or not. We have also given a complete description of these conditional state spaces for the case of quasi-free fermionic states, the fermionic analogues of Gaussian states.
- Kretschmann, D., & Werner, R. F. (2005). Quantum channels with memory. Physical Review A, 72(6), 062323. https://doi.org/10.1103/PhysRevA.72.062323
- Akhalwaya, I., Wouters, J., Fannes, M., & Petruccione, F. (2009). The Algebraic Measure of a Hidden Markov Quantum Memory Channel. AIP Conference Proceedings, 1110(1), 127–130. https://doi.org/10.1063/1.3131288
- Wouters, J., Fannes, M., Akhalwaya, I., & Petruccione, F. (2009). Classical capacity of a qubit depolarizing channel with memory. Physical Review A, 79(4), 042303. https://doi.org/10.1103/PhysRevA.79.042303
- Wouters, J. (2011). Quantum Hidden Markov Chains. [PhD thesis].
- Fannes, M., & Wouters, J. (2009). Correlations in free fermionic states. Journal of Physics A: Mathematical and Theoretical, 42(46), 465308.
- Fannes, M., Nachtergaele, B., & Werner, R. F. (1992). Finitely correlated states on quantum spin chains. Communications in Mathematical Physics, 144(3), 443–490. https://doi.org/10.1007/BF02099178