Quantum Mechanics as Hamilton–Killing Flows on a Statistical Manifold

The mathematical formalism of quantum mechanics is derived or “reconstructed” from more basic considerations of the probability theory and information geometry. The starting point is the recognition that probabilities are central to QM; the formalism of QM is derived as a particular kind of flow on a finite dimensional statistical manifold—a simplex. The cotangent bundle associated to the simplex has a natural symplectic structure and it inherits its own natural metric structure from the information geometry of the underlying simplex. We seek flows that preserve (in the sense of vanishing Lie derivatives) both the symplectic structure (a Hamilton flow) and the metric structure (a Killing flow). The result is a formalism in which the Fubini–Study metric, the linearity of the Schrödinger equation, the emergence of complex numbers, Hilbert spaces and the Born rule are derived rather than postulated.

Observer Dependent Biases of Quantum Randomness

Quantum mechanics (QM) proposes that a quantum system measurement does not register a pre-existing reality but rather establishes reality from the superposition of potential states. Measurement reduces the quantum state according to a probability function, the Born rule, realizing one of the potential states. Consequently, a classical reality is observed. The strict randomness of the measurement outcome is well-documented (and theoretically predicted) and implies a strict indeterminacy in the physical world’s fundamental constituents. Wolfgang Pauli, with Carl Gustav Jung, extended the QM framework to measurement outcomes that are meaningfully related to human observers, providing a psychophysical theory of quantum state reductions. The Pauli-Jung model (PJM) proposes the existence of observer influences on quantum measurement outcomes rooted in the observer’s unconscious mind. The correlations between quantum state reductions and (un)conscious states of observers derived from the PJM and its mathematical reformulation within the model of pragmatic information (MPI) were empirically tested. In all studies, a subliminal priming paradigm was used to induce a biased likelihood for specific quantum measurement outcomes (i.e., a higher probability of positive picture presentations; Studies 1 and 2) or more pronounced oscillations of the evidence than expected by chance for such an effect (Studies 3 and 4). The replicability of these effects was also tested. Although Study 1 found strong initial evidence for such effects, later replications (Studies 2 to 4) showed no deviations from the Born rule. The results thus align with standard QM, arguing against the incompleteness of standard QM in psychophysical settings like those established in the studies. However, although no positive evidence exists for the PJM and the MPI, the data do not entirely falsify the model’s validity.

Obscure Qubits and Membership Amplitudes

We propose a concept of quantum computing which incorporates an additional kind of uncertainty, i.e. vagueness (fuzziness), in a natural way by introducing new entities, obscure qudits (e.g. obscure qubits), which are characterized simultaneously by a quantum probability and by a membership function. To achieve this, a membership amplitude for quantum states is introduced alongside the quantum amplitude. The Born rule is used for the quantum probability only, while the membership function can be computed from the membership amplitudes according to a chosen model. Two different versions of this approach are given here: the “product” obscure qubit, where the resulting amplitude is a product of the quantum amplitude and the membership amplitude, and the “Kronecker” obscure qubit, where quantum and vagueness computations are to be performed independently (i.e. quantum computation alongside truth evaluation). The latter is called a double obscure-quantum computation. In this case, the measurement becomes mixed in the quantum and obscure amplitudes, while the density matrix is not idempotent. The obscure-quantum gates act not in the tensor product of spaces, but in the direct product of quantum Hilbert space and so called membership space which are of different natures and properties. The concept of double (obscure-quantum) entanglement is introduced, and vector and scalar concurrences are proposed, with some examples being given.

Discrete Hilbert space, the Born Rule, and quantum gravity

Quantum gravitational effects suggest a minimal length, or spacetime interval, of order of the Planck length. This in turn suggests that Hilbert space itself may be discrete rather than continuous. One implication is that quantum states with norm below some very small threshold do not exist. The exclusion of what Everett referred to as maverick branches is necessary for the emergence of the Born Rule in no collapse quantum mechanics. We discuss this in the context of quantum gravity, showing that discrete models (such as simplicial or lattice quantum gravity) indeed suggest a discrete Hilbert space with minimum norm. These considerations are related to the ultimate level of fine-graining found in decoherent histories (of spacetime geometry plus matter fields) produced by quantum gravity.