Monotonicity of entropy of sums of quantum random variables
T.J. van de Lisdonk (TU Delft - Applied Sciences)
M.P.T. Caspers – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M. Blaauboer – Mentor (TU Delft - Applied Sciences)
J. Komjáthy – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
M.C. Goorden – Graduation committee member (TU Delft - Applied Sciences)
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Abstract
This thesis studies how von Neumann entropy behaves in two closely related discrete quantum settings: a quantum Central Limit Theorem (qCLT) construction and bosonic mode gate cascades used in quantum optics. The qCLT, originally introduced by Cushen and Hudson [CH71], provides a quantum analogue of the classical Central Limit Theorem, which describes how repeated convolution leads to convergence to a Gaussian state. An extension of this analogy would be the monotonic increase of the von Neumann entropy in this setting, as a similar result was proven in the classical case, and monotonicity of the von Neumann entropy would align with thermodynamic intuition.
The first part of the thesis develops the theoretical background of the qCLT, with emphasis on the Stone-von Neumann theorem, which guarantees a unique representation of the canonical commutation relation. Special attention is paid to an interpretation in which the qCLT evolution is seen as a discrete
form of thermalisation with the environment. The second part focuses on numerical simulations of the qCLT evolutions, where the resulting entropy behaviour is analysed and compared with theoretical expectations.
In addition to the qCLT evolutions, a mathematically related though physically distinct system is considered in the form of bosonic mode gate cascades. Both fixed and randomly varying unitary gates in the cascade are analysed, and convergence towards the expected limiting states is consistently observed.
However, monotonicity of the entropy does not hold in general when the unitary operators representing the gates are chosen randomly, since such unitaries are not restricted to physically structured transformations (unlike beamsplitters, which must satisfy specific physical constraints) and therefore need not
systematically increase entropy, with both entropy-increasing and entropy-decreasing effects occurring.
Overall, the results show a clear distinction between convergence and monotonicity. While convergence to equilibrium appears to be a consistent feature of both models, monotonic entropy behaviour depends much more sensitively on the structure of the underlying unitary dynamics. This suggests that physically motivated constraints on the allowed unitaries are essential if one wants to recover entropy monotonicity in discrete quantum evolutions.