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arXiv:quant-ph/0108076 · Physical Review A

Nonlocal Hamiltonian simulation assisted by local operations and classical communication

Guifré Vidal, J. I. Cirac

Published Aug 27, 2002• 34 citations
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Abstract

Consider a set of N systems and an arbitrary interaction Hamiltonian H that couples them. We investigate the use of local operations and classical communication (LOCC), together with the Hamiltonian H, to simulate a unitary evolution of the N systems according to some other Hamiltonian ${H}^{\ensuremath{'}}.$ First, we show that the most general simulation using H and LOCC can also be achieved, with the same time efficiency, by just interspersing the evolution of H with local unitary manipulations (LU) of each system and a corresponding local ancilla (in a so-called LU+anc. protocol). Thus, the ability to make local measurements and to communicate classical information does not help in nonlocal Hamiltonian simulation. Second, we show that both for the case of two d-level systems $(d>2),$ or for that of a setting with more than two systems $(N>2),$ LU+anc. protocols are more powerful than LU protocols. Therefore local ancillas are a useful resource for nonlocal Hamiltonian simulation. Third, we use results of majorization theory to explicitly solve the problem of optimal simulation of two-qubit Hamiltonians using LU (equivalently, LU+anc., LO, or LOCC).

DOI
10.1103/physreva.66.022315

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Consider a set of N systems and an arbitrary interaction Hamiltonian H that couples them. We investigate the use of local operations and classical communication (LOCC), together with the Hamiltonian H, to simulate a unitary evolution of the N systems according to some other Hamiltonian H\ensuremath′.{H}^{\ensuremath{'}}.H\ensuremath′. First, we show that the most general simulation using H and LOCC can also be achieved, with the same time efficiency, by just interspersing the evolution of H with local unitary manipulations (LU) of each system and a corresponding local ancilla (in a so-called LU+anc. protocol). Thus, the ability to make local measurements and to communicate classical information does not help in nonlocal Hamiltonian simulation. Second, we show that both for the case of two d-level systems (d>2),(d>2),(d>2), or for that of a setting with more than two systems (N>2),(N>2),(N>2), LU+anc. protocols are more powerful than LU protocols. Therefore local ancillas are a useful resource for nonlocal Hamiltonian simulation. Third, we use results of majorization theory to explicitly solve the problem of optimal simulation of two-qubit Hamiltonians using LU (equivalently, LU+anc., LO, or LOCC).
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.
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.
Consider two quantum systems A and B interacting according to a product Hamiltonian H=H/sub A//spl ominus/H/sub B/. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all nonlocal features of a product Hamiltonian - including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians - depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for the entanglement capacity of all product Hamiltonians. Finally, we show how the notion of simulation leads to a natural formulation of measures of the strength of a nonlocal Hamiltonian.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.
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.
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.