Abstract
Entropic uncertainty is a well-known concept to formulate uncertainty relations for continuous variable quantum systems with finitely many degrees of freedom. Typically, the bounds of such relations scale with the number of oscillator modes, preventing a straight-forward generalization to quantum field theories. In this work, we overcome this difficulty by introducing the notion of a functional relative entropy and show that it has a meaningful field theory limit. We present the first entropic uncertainty relation for a scalar quantum field theory and exemplify its behavior by considering few particle excitations and the thermal state. Also, we show that the relation implies the multidimensional Heisenberg uncertainty relation.
Highlights
We present the first entropic uncertainty relation for a scalar quantum field theory and exemplify its behavior by considering few particle excitations and the thermal state
We extend the concept of entropic uncertainty to scalar quantum field theories, for which our motivation is threefold
The notion of distinguishability measured in terms of relative entropy may be considered to be more universal than the notion of missing information measured in terms of entropy. Motivated by these properties of the relative entropy, we have unified discrete and continuous entropic uncertainty relations in Ref. [51], leading to an upper bound for a sum of two relative entropies with respect to maximum entropy model distributions. We extend this idea to free scalar quantum fields, to obtain the relative entropic uncertainty relation given in eq (52), which holds for a collection of oscillators as well as for scalar quantum fields
Summary
The uncertainty principle is one of the most well-known features of quantum mechanics. (1) and (2)), the relative entropic uncertainty relation (52) provides a non-trivial upper bound for a sum of two relative entropies in terms of differences of two-point correlation functions. We will develop the idea of a field-theoretic relative entropic uncertainty relation by means of a free scalar field theory In this case, the field operator and the conjugate momentum field operator fulfill a bosonic commutation relation (analogous to position and momentum in quantum mechanics) and the vacuum state has a Gaussian Schrödinger functional. We derive our relative entropic uncertainty relation and demonstrate its independence of the number of oscillator modes by considering examples, namely excited states and the thermal state, in the field theory limit. We refer to vacuum quantities by using a bar, e.g., F[φ]
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