H ij Entangle- ment flow multipartite systems [1] Numerically computed times assuming saturated rate equations, along with the lower bound (solid line) and “bound on the best bound” (dashed line). For systems in pure states, entangle- ment flow through a particle is limited by its entanglement with the rest of the system. (Previous work has already shown that this does not hold for mixed states, where entanglement can ‘tunnel’ through particles.) In networks of interacting particles, entanglement flow along the network can be described by a set of rate equations, analogous to the Arrhenius equations for chemical reactions. We have used the rate equations to prove bounds on the scaling with chain length of the time required to entangle the ends of a chain. Any system can be viewed as a tri- partite chain: the two systems being entangled, and everything else lumped into one mediating system. An equation analogous to the 3-qubit result holds for the entangled fraction in general tripartite chains:  i j is the entangled fraction of B with the rest. Subtracting 1 rescales it to zero for separable states. In a large multipartite system, lumping particles together hides much of the dynamics. Can we more fully describe entanglement flow in networks of interacting particles? Our inspiration is loosely based on the Arrhenius equations for chemical reactions. A reaction mechanism describes the steps by which reactants are transformed, via successive intermediate compounds, into the final products. The rate at which a compound is produced depends on the amounts of its precursors. The complete reaction is described by a set of coupled rate equations, one for each step in the reaction. We have derived a similar set of rate equations for entanglement. The entangled fraction is also an experimentally motivated quantity. E.g. it determines the fidelity of a teleported state. And it gives bounds on LOCC measures such as the ent. of formation. Until recently, work in entanglement theory concentrated on understanding static properties of entangled quantum states. Although still not fully understood, significant progress has been made in understanding entanglement statics: for example bipartite pure-state entanglement is well understood. This begs the question: what happens when we let the state evolve? Moving from entanglement statics to entanglement dynamics raises many new and interesting questions. We address one of the more fundamental of these: how does entanglement evolve as particles in a multipartite system interact? We establish a quantitative concept of entanglement flow, both through individual particles and along entire networks of inter- acting particles. Surprisingly, we showed in previous work [2] that entanglement can be created between the two end particles without the middle ones ever becoming entangled. However, this can only occur if the system is in a mixed state. For pure states, no entanglement can be created between the end particles if the middle one is not entangled. This can be quantified: if the middle particle is only slightly entangled, entanglement `flows’ slowly: Singlet fraction curves F k ( t ) for saturated rate equations. It can be shown this is the fastest allowable evolution. As an application, the entanglement rate equations can be used to prove bounds on the time T ent required to entangle the end particles in a qubit chain – or more precisely, the scaling of this time with chain length L. We find the following square-root bound: It can also be shown that, up to a constant factor, this is the best bound that can be obtained from the rate equations. Flow through particles 3-qubit chains Introduction Bounds for qubit chains Conclusions [1] quant-ph/ [2] PRL 91, (2003) References T. S. Cubitt F. Verstraete J.I. Cirac Sum is over inter- actions crossing boundary of AB (red in fig.) The entanglement rate equations involve the generalized singlet fraction for qubits a & b embedded in larger systems A & B : The rate equations link the generalized singlet fraction F AB of a set of particles AB with that of the set A ’ B ’ containing all particles interacting directly with AB : The rate at which entanglement is generated depends on the entanglement further back along the network. The dynamics of the complete system is described by a coupled set of rate equations, one for each step in the interaction network. (tr /ab is the partial trace over everything except ab). Rate equations Chemical analogy Entanglement measures are usually defined in the LOCC paradigm: local operations and classical communication should not change the entanglement. However, in a system of interacting particles, it is not clear what classical communication means. Instead, entanglement measures can be defined in the local-unitary paradigm: any change to a state due to local terms in the Hamiltonian should not change the entanglement. The entangled fraction (fidelity) satisfies this: where jÁi is a maximally entangled state. Fidelity & entangled fractions H ab H bc c a b