On Lorentzian symmetries of quantum information

Abstract

A foundational result in relativistic quantum information theory due to Peres, Scudo, and Terno, is that von Neumann entropy is not Lorentz invariant. Motivated by the "It from Qubit" paradigm, here we show that Lorentzian symmetries of quantum information emerge naturally in a pre-spacetime setting, without any reference to external variables such as position or momentum. In particular, we derive the natural action of the restricted Lorentz group $\text{SO}^+(1,3)$ on the internal degrees of freedom of a single qubit from a simple, information-theoretic principle we refer to as preservation of linear entropy. It is then shown that the Lorentz invariance of the linear entropy of a relativistic qubit is a special case of a much more general phenomenon, namely, that any spectral invariant of an operator we term the '$W$-matrix' is an $\text{SL}(2,\mathbb C)^{\otimes n}$ invariant scalar. Consequently, the linear $n$-partite quantum mutual information is shown to be an $\text{SL}(2,\mathbb C)^{\otimes n}$ invariant for all $n$-qubit states. Finally, we show that the correlation function associated with a pair of qubits in the singlet state yields the Minkowski metric on the space of qubit observables, whose symmetry group is the full Lorentz group $\text{SO}(1,3)$. In accordance with the "It from Qubit" paradigm, our results thus establish the natural emergence of relativistic spacetime structure from intrinsic properties of quantum information.