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arXiv:quant-ph/0112124 · Physical Review Letters

Optimal Conversion of Nonlocal Unitary Operations

Wolfgang Dür, Guifré Vidal, J. I. Cirac

Published Jul 15, 2002• 70 citations
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Abstract

We study when a nonlocal unitary operation acting on two d-level systems can probabilistically simulate another one when arbitrary local operations and classical communication are allowed. We provide necessary and sufficient conditions for the simulation to be possible. Probabilistic interconvertability is used to define an equivalence relation between gates. We show that this relation induces a finite number of classes that we identify. In the case of two qubits, two classes of nonlocal operations exist. We show how the representatives of these classes, CNOT and SWAP, can be deterministically converted into any operation of its class and calculate the optimal probability of the reverse process.

DOI
10.1103/physrevlett.89.057901

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We study when a nonlocal unitary operation acting on two d-level systems can probabilistically simulate another one when arbitrary local operations and classical communication are allowed. We provide necessary and sufficient conditions for the simulation to be possible. Probabilistic interconvertability is used to define an equivalence relation between gates. We show that this relation induces a finite number of classes that we identify. In the case of two qubits, two classes of nonlocal operations exist. We show how the representatives of these classes, CNOT and SWAP, can be deterministically converted into any operation of its class and calculate the optimal probability of the reverse process.
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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 $H\ensuremath{\ne}{H}_{A}{+H}_{B}.$ We consider the achievable space of Hamiltonians ${H}^{\ensuremath{'}}$ such that the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}t}$ can be simulated by the interaction H interspersed with local operations. For any dimensions of A and B, and any nonlocal Hamiltonians H and ${H}^{\ensuremath{'}},$ there exists a scale factor s such that for all times t the evolution ${e}^{\ensuremath{-}{\mathrm{iH}}^{\ensuremath{'}}\mathrm{st}}$ can be simulated by H acting for time t interspersed with local operations. For two-qubit Hamiltonians H and ${H}^{\ensuremath{'}},$ 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 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 Hnot equalH(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 two-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 two-qubit Hamiltonians.
Suppose that two distant parties Alice and Bob share an entangled state ρAB\rho_{AB}ρAB​, and they want to exchange the subsystems of ρAB\rho_{AB}ρAB​ by local operations and classical communication (LOCC). In general, this LOCC task (i.e. the LOCC transformation of ρAB→VρABV\rho_{AB} \to V\rho_{AB} VρAB​→VρAB​V with VVV being a swap operator) is impossible deterministically, but becomes possible probabilistically. In this paper, we study how the optimal probability is related to the amount of entanglement in the framework of positive partial transposed (PPT) operations, and numerically show a remarkable class of states whose probability is the smallest among every state in two quantum bits.
We present a method of optimizing recently designed protocols for implementing an arbitrary nonlocal unitary gate acting on a bipartite system. These protocols use only local operations and classical communication with the assistance of entanglement, and they are deterministic while also being ``one-shot,'' in that they use only one copy of an entangled resource state. The optimization minimizes the amount of entanglement needed, and also the amount of classical communication, and it is often the case that less of each of these resources is needed than with an alternative protocol using two-way teleportation.