Stochastic Mechanics as a Gauge Theory

C. Albanese

We show that non-relativistic Quantum Mechanics can be faithfully represented in terms of a classical diffusion process endowed with a gauge symmetry of group Z_4. The representation is based on a quantization condition for the realized action along paths. A lattice
regularization is introduced to make rigorous sense of the construction and then removed. Quantum mechanics is recovered in the continuum limit and the full U(1) gauge group symmetry of electro-magnetism appears. Anti-particle representations emerge naturally, albeit the context is non-relativistic. Quantum density matrices are obtained by averaging classical
probability distributions over phase-action variables. We find that quantum conditioning can be described in classical terms but not through the standard notion of sub sigma−algebras. Delicate restrictions arise by the constraint that we are only interested in the algebra of gauge invariant random variables. We conclude that Quantum Mechanics is equivalent to a theory of gauge invariant classical stochastic processes we call Stochastic Mechanics.

Mirror inversion of quantum states in linear registers

C. Albanese, Matthias Christandl, Nilanjana Datta and Artur Ekert

Transfer of data in linear quantum registers can be significantly simplified with pre-engineered but not dynamically controlled inter-qubit couplings. We show how to implement a mirror inversion of a quantum state of qubits with respect to the centre of the register. Our construction is especially appealing as it requires no dynamical control over individual inter-qubit interactions. If, however, the individual control is available then the mirror inversion operation can be performed on any substring of qubits in the register. In this case a sequence of mirror inversions can eciently generate any permutation of a quantum state of the involved qubits.

Classification of Solvable Mirror Periodic Chains

C. Albanese and S. Lawi

We present a classification scheme for mirror periodic quantum spin chains with nearest neighbor couplings whose eigenstates can be expressed in analytically closed form in terms of hypergeometric polynomials. These chains of arbitrary finite length exhibit a strong state transfer property, according to which the mirror image of a state is periodically reconstituted. We also construct their continuous space limit using the limit relations between hypergeometric polynomials in the Askey scheme.

Time Quantization and q-Deformations

C. Albanese and S. Lawi

We extend to quantum mechanics the technique of stochastic ubordination, by means of which one can express any semi-martingale as a time-changed Brownian motion. As examples, we considered two ersions of the q-deformed Harmonic oscillator in both ordinary and imaginary time and show how these various cases can be understood as different patterns of time quantization rules.