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arXiv:quant-ph/0101107

Probabilistic nonlocal gate operation via imperfect entanglement

Jingak Jang, Jinhyoung Lee, M. S. Kim, Y.-J. Park

Published Jan 22, 2001• 1 citation
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Abstract

Nonlocal gate operation is based on sharing an ancillary pair of qubits in perfect entanglement. When the ancillary pair are partially entangled, the efficiency of the gate operation drops. Using general transformations, we devise probabilistic nonlocal gates, which perform the nonlocal operation conclusively when the ancillary pair are only partially entangled. We show that a controlled purification protocol can be implemented by the probabilistic nonlocal operation.

DOI
10.48550/arxiv.quant-ph/0101107

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Nonlocal gate operation is based on sharing an ancillary pair of qubits in perfect entanglement. When the ancillary pair are partially entangled, the efficiency of the gate operation drops. Using general transformations, we devise probabilistic nonlocal gates, which perform the nonlocal operation conclusively when the ancillary pair are only partially entangled. We show that a controlled purification protocol can be implemented by the probabilistic nonlocal operation.
We propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a purification protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We propose a protocol for implementation of nonlocal CNOT operation using a partially entangled channel and show that when partially entangled pairs are taken as quantum channels, the nonlocal CNOT operation can be implemented probabilistically by introducing a collective unitary transformation. The required resources for implementation of the nonlocal CNOT operation in this case are discussed. We also point out that the nonlocal CNOT operation can be used as a puritlcation protocol to concentrate entanglement from an ensemble of partially entangled particles into a subensemble of maximally entanglement ones.
We show how a nonlocal and open-target destination quantum controlled-Not （CNOT） gate can be probabilistically implemented by using partially entangled pairs of particles. We first investigate the controlled implementation of a nonlocal and three-target destination CNOT gate using three partially entangled pairs, and then generalize the scheme to the case of N-target destination. In this scheme, Alice’s local generalized measurement described by a positive operator valued measurement （POVM） plays a key role. We construct the required POVM. It is worth noting that deterministic and exact implementation of a nonlocal CNOT gate can be realized using partially entangled pairs.
We provide several applications of a previously introduced isomorphism between physical operations acting on two systems and entangled states [Phys. Rev. Lett. 86, 544 (2001)]. We show (i) how to implement (weakly) nonlocal two-qubit unitary operations with a small amount of entanglement; (ii) that a known, noisy, nonlocal unitary operation as well as an unknown, noisy, local unitary operation can be purified; (iii) how to perform the tomography of arbitrary, unknown, nonlocal operations; (iv) that a set of local unitary operations as well as a set of nonlocal unitary operations can be stored and compressed; and (v) how to implement probabilistically two-qubit gates for photons. We also show how to compress a set of bipartite entangled states locally, as well as how to implement certain nonlocal measurements using a small amount of entanglement. Finally, we generalize some of our results to multiparty systems.
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>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 $\ensuremath{\xi}\ensuremath{\leqslant}0.353$. 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.
Remote implementation of nonlocal quantum gates is one of the central tasks in quantum communication. Researchers devote much interest in remote implementation of nonlocal quantum gates via quantum entangled channel. Remote implementation protocols of two-qubit controlled-not gate, three-qubit Toffoli gate have been presented. We present two schemes for remote implementation of nonlocal Toffoli gate via partially entangled quantum channel against collective noise. The first is used to remote implementing of nonlocal Toffoli gate via partially entangled channel against collective-dephasing noise, and the second is used to remote implementing of nonlocal Toffoli gate against collective-rotation noise. The schemes are tolerant of collective-dephasing noise and collective-rotating noise. The agents can implement the Toffoli gate nonlocally with partially entangled quantum channel by introducing an auxiliary qubit and performing general evolution. The schemes are more convenient in application since they only requires partially entangled quantum states for nonlocal implementation of quantum operations.