Spontaneously interacting qubits from Gauss-Bonnet

Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which "know about qubits" (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems.Comment: 29 pages, 7 figure

Classifying the non-time-local and entangling dynamics of an open qubit system

Abstract We study families of dynamical maps generated from interactions with varying degrees of symmetry. For a family of time-independent Hamiltonians, we demonstrate the relationship between symmetry, strong-coupling, perfect entanglers, non-Markovian features, and non-time-locality. We show that by perturbing the initial environment state, effective time-local descriptions can be obtained that are non-singular yet capture essential non-unitary features of the reduced dynamics. We then consider a time-dependent Hamiltonian that changes the degree of symmetry by activating a dormant degree of freedom. In this example we find that the one-qubit reduced dynamics changes dramatically. These results can inform the construction of effective theories of open systems when the larger system dynamics is unknown

Classifying the non-Markovian, non-time-local, and entangling dynamics of an open quantum system

We study the dynamical map and master equation describing the evolution of one qubit coupled to another via a family of Hamiltonians with a parity symmetry. We classify the structure of the dynamical map and the non-Markovian and non-time-local aspects of the single-qubit dynamics according to the degree of symmetry and coupling strength in the full Hamiltonian, as well as the initial state of the environment qubit. We demonstrate the relationship between strong-coupling, perfect entanglers, non-Markovian features, and non-time-locality in this simple system. We show that by perturbing the initial environment state, effective time-local descriptions can be obtained that are non-singular however capture essential non-unitary features of the reduced dynamics. These results can inform the construction of effective theories of open systems when the larger system dynamics is unknown.Comment: 30 pages, 5 figur

Spontaneously interacting qubits from Gauss-Bonnet

Abstract Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which “know about qubits” (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems