Quantum theory does not provide a unique definition for the joint probability of two noncommuting observables, which is the next important question after the Born's probability for a single observable. Instead, various definitions were suggested, e.g. via quasi-probabilities or via hidden-variable theories. After reviewing open issues of the joint probability, we relate it to quantum imprecise probabilities, which are non-contextual and are consistent with all constraints expected from a quantum probability. We study two non-commuting observables in a two-dimensional Hilbert space and show that there is no precise joint probability that applies for any quantum state and is consistent with imprecise probabilities. This contrasts to theorems by Bell and Kochen-Specker that exclude joint probabilities for more than two non-commuting observables, in Hilbert space with dimension larger than two. If measurement contexts are included into the definition, joint probabilities are not anymore excluded, but they are still constrained by imprecise probabilities.

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Coherent population trapping is a well-known quantum phenomenon in a driven Λ system, with many applications across quantum optics. However, when a stochastic bath is present in addition to vacuum noise, the observed trapping is no longer perfect. Here we derive a time-convolutionless master equation describing the equilibration of the Λ system in the presence of additional temporally correlated classical noise, with an unknown decay parameter. Our simulations show a one-to-one correspondence between the decay parameter and the depth of the characteristic dip in the photoluminescence spectrum, thereby enabling the unknown parameter to be estimated from the observed spectra. We apply our analysis to the problem of qubit state initialization in a Λ system via dark states and show how the stochastic bath affects the fidelity of such initialization as a function of the desired dark-state amplitudes. We show that an optimum choice of Rabi frequencies is possible.

a) PACS 03.65.-w -Quantum mechanics PACS 03.67.-a -Quantum informationAbstract -Quantum non-locality is normally defined via violations of Bell's inequalities that exclude certain classical hidden variable theories from explaining quantum correlations. Another definition of non-locality refers to the wave-function collapse thereby one can prepare a quantum state from arbitrary far away. In both cases one can debate on whether non-locality is a real physical phenomenon, e.g. one can employ formulations of quantum mechanics that does not use collapse, or one can simply refrain from explaining quantum correlations via classical hidden variables. Here we point out that there is a non-local effect within quantum mechanics, i.e. without involving hidden variables or collapse. This effect is seen via imprecise (i.e. interval-valued) joint probability of two observables, which replaces the ill-defined notion of the precise joint probability for non-commuting observables. It is consistent with all requirements for the joint probability, e.g. those for commuting observales. The non-locality amounts to a fact that (in a two-particle system) the joint imprecise probability of non-commuting two-particle observables (i.e. tensor product of single-particle observables) does not factorize into single-particle contributions, even for uncorrelated states of the two-particle system. The factorization is recovered for a less precise (i.e. the one involving a wider interval) joint probability. This approach to non-locality reconciles it with locality, since the latter emerges as a less precise description.