We develop a Liouville perturbation theory for weakly driven and weakly open quantum systems in situations when the unperturbed system has a number of conservations laws. If the perturbation violates the conservation laws, it drives the system to a new steady state which can be approximately but efficiently described by a (generalized) Gibbs ensemble characterized by one Lagrange parameter for each conservation law. The value of those has to be determined from rate equations for conserved quantities. Remarkably, even weak perturbations can lead to large responses of conserved quantities. We present a perturbative expansion of the steady state density matrix; first we give the condition that fixes the zeroth-order expression (Lagrange parameters) and then determine the higher-order corrections via projections of the Liouvillian. The formalism can be applied to a wide range of problems including two-temperature models for electron-phonon systems, Bose condensates of excitons or photons, or weakly perturbed integrable models. We test our formalism by studying interacting fermions coupled to nonthermal reservoirs, approximately described by a Boltzmann equation.
I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates the Bell’s inequalities. Taken together with the Bell’s theorem, this result has far reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden variable theories are incompatible with fundamental symmetries and conservation laws. Bell’s inequalities can be obeyed only by violating a conservation law. The implications for experiments on Bell’s inequalities are obvious. The result provides new insight regarding entanglement, and its measures. PACS Numbers: 03.65.Ta, 03.65.Ud The main result of this paper is the proof that a general physical theory of correlations of two-valued spin projections violate the Bell’s inequalities if the theory satisfies the conservation law for angular momentum on the average. The proof is applicable to local hidden variable theories, as well as
We develop a general framework for the statistics of mixed-dimensional excitations subject to intertwined conservation laws, extending the familiar Fermi statistics with conserved particle number. We define statistics microscopically through a \emph{hopping-operator algebra}: a local operator subalgebra (LOsA) generated by operators that locally move or deform excitations while preserving the conservation law. Nontrivial statistics arise when this subalgebra is nontrivial.
We first focus on LOsAs that encode \emph{pointed} conservation laws. These give rise to invertible excitations, whose fusion rules are exactly those of the symmetry defects of a higher group $\cG$. For such $\cG$-conserved excitations in
d-dimensional space, we show that the corresponding LOsA -- and hence the statistics it defines -- is classified by a cohomology class $[ω] \in H^{d+2}(B\cG;\R/\Z)$, where changing
[ω] by a coboundary corresponds merely to a rephasing of the local operators. We further provide a holographic realization: excitations with this prescribed conservation law and statistics live on the boundary of a $\cG$ higher-group gauge theory in
(d+1)-dimensional space, twisted by
[ω].
More generally, non-pointed conservation laws and the associated statistics of non-invertible excitations are defined by a pair: a LOsA together with its excitation-complex representation. This is equivalent to the pair consisting of a LOsA and its Hilbert-space representation, which is the data defining a generalized symmetry. Consequently, non-pointed conservation laws and their statistics in
d-dimensional space are classified by fusion
d-categories, just as generalized symmetries are. The higher-group results above are the fully-pointed special cases of this more general classification.
I derive the correlation function for a general theory of two-valued spin variables that satisfy the fundamental conservation law of angular momentum. The unique theory-independent correlation function is identical to the quantum-mechanical correlation function. I prove that any theory of correlations of such discrete variables satisfying the fundamental conservation law of angular momentum violates Bell's inequalities. Taken together with Bell's theorem, this result has far-reaching implications. No theory satisfying Einstein locality, reality in the EPR-Bell sense, and the validity of the conservation law can be constructed. Therefore, all local hidden-variable theories are incompatible with fundamental symmetries and conservation laws.