Logical pre- and post-selection paradoxes are proofs of contextuality

If a quantum system is prepared and later post-selected in certain states, "paradoxical" predictions for intermediate measurements can be obtained. This is the case both when the intermediate measurement is strong, i.e. a projective measurement with Luders-von Neumann update rule, or with weak measurements where they show up in anomalous weak values. Leifer and Spekkens [quant-ph/0412178] identified a striking class of such paradoxes, known as logical pre- and post-selection paradoxes, and showed that they are indirectly connected with contextuality. By analysing the measurement-disturbance required in models of these phenomena, we find that the strong measurement version of logical pre- and post-selection paradoxes actually constitute a direct manifestation of quantum contextuality. The proof hinges on under-appreciated features of the paradoxes. In particular, we show by example that it is not possible to prove contextuality without Luders-von Neumann updates for the intermediate measurements, nonorthogonal pre- and post-selection, and 0/1 probabilities for the intermediate measurements. Since one of us has recently shown that anomalous weak values are also a direct manifestation of contextuality [arXiv:1409.1535], we now know that this is true for both realizations of logical pre- and post-selection paradoxes.Comment: In Proceedings QPL 2015, arXiv:1511.0118

Quantum Dynamics, Minkowski-Hilbert space, and A Quantum Stochastic Duhamel Principle

In this paper we shall re-visit the well-known Schr\"odinger and Lindblad dynamics of quantum mechanics. However, these equations may be realized as the consequence of a more general, underlying dynamical process. In both cases we shall see that the evolution of a quantum state $P_\psi=\varrho(0)$ has the not so well-known pseudo-quadratic form $\partial_t\varrho(t)=\mathbf{V}^\star\varrho(t)\mathbf{V}$ where $\mathbf{V}$ is a vector operator in a complex Minkowski space and the pseudo-adjoint $\mathbf{V}^\star$ is induced by the Minkowski metric $\boldsymbol{\eta}$. The interesting thing about this formalism is that its derivation has very deep roots in a new understanding of the differential calculus of time. This Minkowski-Hilbert representation of quantum dynamics is called the \emph{Belavkin Formalism}; a beautiful, but not well understood theory of mathematical physics that understands that both deterministic and stochastic dynamics may be `unraveled' in a second-quantized Minkowski space. Working in such a space provided the author with the means to construct a QS (quantum stochastic) Duhamel principle and known applications to a Schr\"odinger dynamics perturbed by a continual measurement process are considered. What is not known, but presented here, is the role of the Lorentz transform in quantum measurement, and the appearance of Riemannian geometry in quantum measurement is also discussed

The Stochastic Representation of Hamiltonian Dynamics and The Quantization of Time

Here it is shown that the unitary dynamics of a quantum object may be obtained as the conditional expectation of a counting process of object-clock interactions. Such a stochastic process arises from the quantization of the clock, and this is derived naturally from the matrix-algebra representation of the nilpotent Newton-Leibniz time differential [Belavkin]. It is observed that this condition expectation is a rigorous formulation of the Feynman Path Integral.Comment: 21 page