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arXiv:quant-ph/0303097 · IEEE Transactions on Information Theory

Reversible Simulation of Bipartite Product Hamiltonians

Andrew M. Childs, Debbie Leung, Guifré Vidal

Published Jun 1, 2004• 19 citations
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Abstract

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.

DOI
10.1109/tit.2004.828069

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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.
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),(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 $H\ensuremath{\ne}{H}_{A}{+H}_{B}.$ 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.
We consider interactions as bidirectional channels, and investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted 1-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give bounds on other capacities and discuss their additivity, along with other open questions.
We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and non- local unitary gates to generate entanglement and transmit clas- sical information. We give analytic expressions for the entangle- ment generating capacity and entanglement-assisted one-way clas- sical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding -shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.