This thesis has examined and extended in intricate detail some well-known models from theoretical physics, to help bring some further information to the debate regarding the quantum-classical transition. Throughout the develop- ments we have been careful to include frequent indicators of the experimental accessibility of results, in recognition of the importance that such corrobora- tion has for scientific advancement. The discussion sections of each chapter provide several suggestions for practical applications and further work, some of which we will summarise here and provide some further comment.
The emergence of differences between the tripartite EPR-typeη-states and the GHZ NOPA-type states in Chapter 2 suggests the possibility that these may translate into experimental differences. The theory that led to the develop- ment of the tripartiteη-states may be extended with relative ease to include a whole family of such potentially interesting states. Firstly, note that the pre- sentation discussed the canonically conjugate pair of difference in position and total momentum. A parallel exposition may easily be presented for the alter- native starting point of difference in momentum and total position, along with
114 5.2. DISCUSSION AND FURTHER WORK
the tripartite counterpart to this example. Moreover, the η-state regularisa- tion parameters can be exchanged for any other regularisation parameter that preserves the state symmetry, and can include cases where the regularisation weighting of each mode is considered independently (i.e. three regularisation parameters for the tripartite state). In Subsection 2.3.2 we examined a few clear choices for phase space parameters to extremise the CHSH inequalities using theη-states, but a fuller investigation could document the behaviour of these inequalities for the entire parameter space. A similar analysis can also be conducted with the N-partite state generalisation.
We have chosen to present in this thesis the CHSH violation of the Gaussian limitη= 0 for the EPR-type states, as this is required for current experimental protocols. However, it may well be of theoretical interest to examine the full non-Gaussian expressions in more detail, and they may in the future provide some experimentally accessible information. To further develop the practical applicability of the current results it will be instructive to formalise the re- lationship between the Bell-operator notation for the CHSH inequalities and the variance calculations used in quantum optics, as well as the experimental interpretation of the new regularisation parameters and their limits.
In the thesis we used the EPR example with CHSH inequality violation to highlight the inconsistency in current fundamental theory. In terms of the interpretation of this inequality violation, we mentioned in the Introduction that it is regarded as a violation of locality, and that there is some contro- versy over what this means in practical terms, with the distinction between locality and causality. The EPR experiment, which gives rise to correlations repeatedly confirmed by experiment, is simply a mathematical consequence of entangled states, and in itself offers no explanation of its relationship to classical environments or special relativity. Nevertheless the example is clear in its confirmation of a paradox in fundamental physical theory by juxtapos- ing the clear understanding of the mathematical structure of entangled states and the strange consequences for classical interpretation – that locality can be violated.
Having seen the effect of quantum structures on classical interpretation, we continued the investigation into the quantum-classical relationship by study- ing the effects of a classical environment interacting with a quantum system in
Chapters 3 and 4. In the process we demonstrated the utility of the construc- tive bosonisation method for relating fermion gas-impurity models to their quantum dissipative system counterpart, and exposed the intricacies of the technique. It is worth noting that while we applied the bosonisation proce- dure to Hamiltonians in this thesis the method applies equally well to the bosonisation of states, and it may be informative to bosonise the fermionic bath from the Kondo model, for example.
The re-analysis of the well-known XY Z-type model in Subsection 3.3.2 us- ing the detailed bosonisation procedure showed the model corresponds to an exactly solvable, extended two-level system with the upper and lower levels having infinite degeneracy. This may facilitate new developments for quantum random walks, where each step of the walk is interpreted as a jump to a neigh- bouring site on the lattice. Bethe Ansatz methods may provide new analytic insight into the walker behaviour, and further investigations may reveal the details of the influence of the oscillator bath on the walker performance. The analysis of an Anisotropic Coqblin-Schrieffer model in Section 3.4 led to a new exactly solvable three-level dissipative system. It might be possible to map the bosonised ACS model to other dissipative systems through alterna- tive standard R-matrices. Since the ACS Hamiltonian is only one of many possible Fermi-gas starting points for the bosonisation procedure, it could be expected that the same method might reveal more exactly solvable dissipa- tive systems. The range of applications and importance of such models in condensed matter theory makes this an exciting prospect indeed. The theory for the three-level system discussed herein applies to any triatomic triple well potential, such as ammonia, the methyl −CH3 and Bose-Einstein condensate atomic transistors, to mention only a few. Due to the many applications of models similar to the bosonised ACS model, there are a wealth of different ap- proaches to their analysis in the literature. The ACS model deserves a rigorous contextualisation within this greater body of research, but this is beyond the scope of this thesis. A review of the models and the relationship between their mathematical descriptions would require a detailed analysis, but the yield in terms of interpretational clarity may well make the task worthwhile.
It was demonstrated in Subsection 3.4.1 that the ACS model is exactly solv- able. The consequence is that one of the most important extensions to the
116 5.2. DISCUSSION AND FURTHER WORK
work on the ACS model is finding its exact solution by the Bethe Ansatz. Not only does this enable the calculation of a range of useful dynamical and thermodynamical quantities such as magnetic susceptibility and specific heat, but also the exact values of entropy calculations through the derivatives of the energy eigenvalues as indicated in Chapter 4. For the cases where an exact solution is not known, we showed in Section 4.2 that useful entropy measures may be found nevertheless, using a variational approach. We gave clear procedures for calculating these, and calculations of the reduced den- sity matrices showed a more subtle dependency on the particular states of the impurity and bath than was apparent in the spin-12 Kondo case. Explicit numerical evaluation of the entropy measures can be done by finding the val- ues of the variational parameters involved. Moreover, the investigation can be extended by considering alternative possible ground states for the model and finding the corresponding entropy measures. A most interesting extension for both the variational and Feynman-Hellmann approaches is the calculation of entanglement measures, rather than simply entanglement criteria from the entropy. However, as mentioned in Section 4.4, the validity of any particular entropy measure for dealing with the structure of entangled states should also be considered in more detail.
So far we have mentioned a range of generalisations and suggestions for further work for each of the investigations covered in this thesis. The overarching topic of this thesis – the quantum-classical transition – is of course so wide that we could never hope to touch on every facet in one thesis. We have only very lightly introduced the concept of decoherence, and discussed its relationship to dissipation, but it is clear that a fuller investigation of its relationship with all the parameters in the models is of crucial importance to the topic. In that regard, one might also want to consider the relationship between all four of the canonical system-environment models used to discuss dissipation and decoher- ence. The four include spin-boson and spin-spin models as discussed herein, as well as boson-boson models (quantum Brownian motion) and boson-spin mod- els [111]. Much work has been done in formalising the relationship between the Spin-Boson and Kondo models, but the details of this fuller canonical set of models may contain further insights.
els relating to the quantum-classical interface. The investigation has revealed new insights and the development of new models, and we have suggested sev- eral possibilities for practical applications. Particular focus has been on high- lighting the tension between quantum and classical theory and establishing the structure of entangled states, using quantum system-environment models to reveal details of their interaction, and providing a means for investigating their entanglement relationship. The further work suggested in this chapter – in particular solving the Bethe Ansatz for the ACS model, finding explicit and exact values for the entanglement criteria, and extending the investigation to precise entanglement measures – shows great promise for further develop- ing the examination of the quantum-classical transition. As such the work presented herein provides a firm foundation for the re-examination of this re- lationship more generally.