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arXiv:quant-ph/0501107 · Physical Review A

Probabilistic implementation of a nonlocal operation using a nonmaximally entangled state

Lin Chen, Yi-Xin Chen

Published May 17, 2005• 27 citations
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Abstract

We develop the probabilistic implementation of a nonlocal gate $\mathrm{exp}[i\ensuremath{\xi}{\ensuremath{\sigma}}_{{n}_{A}}{\ensuremath{\sigma}}_{{n}_{B}}]$ and $\ensuremath{\xi}∊[0,(\ensuremath{\pi}∕4)]$, by using a single nonmaximally entangled state. We prove that nonlocal gates can be implemented with a fidelity of $>79.3%$ and a consumption of $<0.969\phantom{\rule{0.3em}{0ex}}\mathrm{ebits}$ and two classical bits, when $\ensuremath{\xi}\ensuremath{\leqslant}0.353$. This provides a higher bound for the feasible operation compared to the former techniques. Besides, gates with $\ensuremath{\xi}\ensuremath{\geqslant}0.353$ can be implemented with the probability 79.3% and a consumption of 0.969 ebits, which is the same efficiency as the distillation-based protocol, while our method saves extra classical resources. Gates with $\ensuremath{\xi}\ensuremath{\rightarrow}0$ can be implemented with nearly unit probability and a small entanglement. We also generalize some applications to the multiple system, where we find it is possible to implement certain nonlocal gates between many nonentangled partners using a nonmaximally multiple entangled state.

DOI
10.1103/physreva.71.054302

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We develop the probabilistic implementation of a nonlocal gate exp[i\ensuremathξ\ensuremathσnA\ensuremathσnB]\mathrm{exp}[i\ensuremath{\xi}{\ensuremath{\sigma}}_{{n}_{A}}{\ensuremath{\sigma}}_{{n}_{B}}]exp[i\ensuremathξ\ensuremathσnA​​\ensuremathσnB​​] and \ensuremathξ∊[0,(\ensuremathπ∕4)]\ensuremath{\xi}∊[0,(\ensuremath{\pi}∕4)]\ensuremathξ∊[0,(\ensuremathπ∕4)], by using a single nonmaximally entangled state. We prove that nonlocal gates can be implemented with a fidelity of >79.3>79.3%>79.3 and a consumption of <0.969ebits<0.969\phantom{\rule{0.3em}{0ex}}\mathrm{ebits}<0.969ebits and two classical bits, when \ensuremathξ\ensuremath⩽0.353\ensuremath{\xi}\ensuremath{\leqslant}0.353\ensuremathξ\ensuremath⩽0.353. This provides a higher bound for the feasible operation compared to the former techniques. Besides, gates with \ensuremathξ\ensuremath⩾0.353\ensuremath{\xi}\ensuremath{\geqslant}0.353\ensuremathξ\ensuremath⩾0.353 can be implemented with the probability 79.3% and a consumption of 0.969 ebits, which is the same efficiency as the distillation-based protocol, while our method saves extra classical resources. Gates with \ensuremathξ\ensuremath→0\ensuremath{\xi}\ensuremath{\rightarrow}0\ensuremathξ\ensuremath→0 can be implemented with nearly unit probability and a small entanglement. We also generalize some applications to the multiple system, where we find it is possible to implement certain nonlocal gates between many nonentangled partners using a nonmaximally multiple entangled state.
We propose and study a method for using nonmaximally entangled states to probabilistically implement desired nonlocal gates. Unlike distillation-based protocols, this method does not generate a maximally entangled state at intermediate stages of the process. As a consequence, the method becomes more efficient at a certain range of parameters. Gates of the form $\mathrm{exp}[i\ensuremath{\xi}{\ensuremath{\sigma}}_{{n}_{A}}{\ensuremath{\sigma}}_{{n}_{B}}]$ with $\ensuremath{\xi}⪡1$ can be implemented with nearly unit probability and with vanishingly small entanglement, while for the distillation-based method, the gate is produced with a vanishing success probability. We also derive an upper bound to the optimal success probability and show that, in the small-entanglement limit, the bound is tight.
We propose a method to probabilistically implement a nonlocal operation, exp[iξ U A U B ], between two distant qutrits A and B , where ξ∊[0,2π] and U A , U B are local unitary and Hermitian operations for qutrits A and B respectively. The consumptions of resource for one performance of the method are a single non-maximally entangled qutrit state and 1-trit classical communication. For a given ξ, the successful probability of the method depends on the forms of both entanglement resource and Bob's partial-measurement basis. We systematically discuss the optimal successful probabilities and their corresponding conditions for three cases: adjustable entanglement resource, adjustable partial-measurement basis, adjustable entanglement resource and partial-measurement basis. It is straightforward to generalize the method for producing nonlocal unitary operations between any two N -level systems.
We propose a scheme for the implementation of nonlocal quantum swap operation on two spatially separated entangled pairs and we show that the operation can swap two qubits of these entangled pairs. We discuss the resources of the entangled qubits and classical communication bits required for the optimal implementation of the nonlocal quantum swap operation. We also put forward a scheme for probabilistic implementation of nonlocal swap operation via a nonmaximally entangled quantum channel. The probability of a successful nonlocal swap operation is obtained by introducing a collective unitary transformation.
We propose a scheme for the implementation of nonlocal quantum swap operation on two spatially separated entangled pairs and we show that the operation can swap two qubits of these entangled pairs.We discuss the resources of the entangled qubits and classical communication bits required for the optimal implementation of the nonlocal quantum swap operation.We also put forward a scheme for probabilistic implementation of nonlocal swap operation via a nonmaximally entangled quantum channel.The probability of a successful nonlocal swap operation is obtained by introducing a collective unitary transformation.
We show how a non-local quantum controlled-NOT (CNOT) gate with multiple targets can be implemented with unit fidelity and unit probability. The explicit quantum circuit for implementing the operation is presented. Two schemes for probabilistic implementing the operation via partially entangled quantum channels with unit fidelity are put forward. The overall physical resources required for accomplishing these schemes are different, and the successful implementation probabilities are also different.