Guifré Vidal

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement or classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems A and B interact by a nonlocal Hamiltonian $H\ne H_A{+H}_B.$ We consider the achievable space of Hamiltonians $H^{\prime }$ such that the evolution $e^{-{\mathrm{iH}}^{\prime }t}$ can be simulated by the interaction H interspersed with local operations. For any dimensions of A and B, and any nonlocal Hamiltonians H and $H^{\prime },$ there exists a scale factor s such that for all times t the evolution $e^{-{\mathrm{iH}}^{\prime }\mathrm{st}}$ can be simulated by H acting for time t interspersed with local operations. For two-qubit Hamiltonians H and $H^{\prime },$ we calculate the optimal s and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of two-qubit Hamiltonians.

We consider the simulation of the dynamics of one nonlocal Hamiltonian by another, allowing arbitrary local resources but no entanglement nor classical communication. We characterize notions of simulation, and proceed to focus on deterministic simulation involving one copy of the system. More specifically, two otherwise isolated systems $A$ and $B$ interact by a nonlocal Hamiltonian $H \neq H_A+H_B$. We consider the achievable space of Hamiltonians $H'$ such that the evolution $e^{-iH't}$ can be simulated by the interaction $H$ interspersed with local operations. For any dimensions of $A$ and $B$, and any nonlocal Hamiltonians $H$ and $H'$, there exists a scale factor $s$ such that for all times $t$ the evolution $e^{-iH'st}$ can be simulated by $H$ acting for time $t$ interspersed with local operations. For 2-qubit Hamiltonians $H$ and $H'$, we calculate the optimal $s$ and give protocols achieving it. The optimal protocols do not require local ancillas, and can be understood geometrically in terms of a polyhedron defined by a partial order on the set of 2-qubit Hamiltonians.

We consider the problem of preparing a single copy of an arbitrary two-qubit mixed state $\rho ,$ starting from an entangled pure state $\psi$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure-state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.

We consider low-rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary M and N $(M<~N),$ and with a positive partial transpose (PPT) ${\varrho }^{T_A}>~0.$ For rank $r\left(\varrho \right)<~N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and ${\varrho }^{T_A}$ satisfies $r\left(\varrho \right)+r\left({\varrho }^{T_A}\right)<~2MN-M-N+2.\mathrm{}$ This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.

In this note we consider the problem of preparing a {\em single} copy of an arbitrary two-qubit mixed state $\rho$ starting from an entangled pure state $\psi$ and using only local operations assisted with classical communication. We present an analytical expression for the minimal amount of pure state entanglement required, and describe the corresponding local strategy. We also examine optimal probabilistic generalizations of the previous process.