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arXiv:quant-ph/0005061 · Physical Review A

Quantum remote control: Teleportation of unitary operations

Susana F. Huelga, Joan A. Vaccaro, Anthony Chefles, Martin Bodo Plenio

Published Mar 15, 2001• 275 citations
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Abstract

We consider the implementation of an arbitrary unitary operation U upon a distant quantum system. This teleportation of U can be viewed as quantum remote control. We investigate protocols that achieve this using local operations, classical communication, and shared entanglement. Lower bounds on the necessary entanglement and classical communication are determined using causality and the linearity of quantum mechanics. We examine in particular detail the resources required if the remote control is to be implemented as a classical black box. Under these circumstances, we prove that the required resources are, necessarily, those needed for implementation by bidirectional state teleportation.

DOI
10.1103/physreva.63.042303

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We consider the implementation of an arbitrary unitary operation U upon a distant quantum system. This teleportation of U can be viewed as quantum remote control. We investigate protocols that achieve this using local operations, classical communication, and shared entanglement. Lower bounds on the necessary entanglement and classical communication are determined using causality and the linearity of quantum mechanics. We examine in particular detail the resources required if the remote control is to be implemented as a classical black box. Under these circumstances, we prove that the required resources are, necessarily, those needed for implementation by bidirectional state teleportation.
We study the remote implementation of a unitary transformation on the state of a qubit. We show the existence of nontrivial protocols (i.e., using less resources than bidirectional state teleportation) that allow the perfect remote implementation of certain continuous sets of quantum operations. We prove that, up to a local change of basis, only two subsets exist that can be implemented remotely with a nontrivial protocol: Arbitrary rotations around a fixed direction $\stackrel{\ensuremath{\rightarrow}}{n}$ and a $\ensuremath{\pi}$ rotation about an arbitrary direction lying in a plane orthogonal to $\stackrel{\ensuremath{\rightarrow}}{n}.$ The former operations effectively constitute the teleportation of arbitrary angles. The overall classical information and distributed entanglement cost required for the remote implementation depends on whether it is known, a priori, in which of the two teleportable subsets the transformation belongs. If it is known, the optimal protocol consumes one e-bit of entanglement and one c-bit in each direction. If it is not known in which subset the transformation belongs, two e-bits of entanglement need to be consumed and the classical channel becomes asymmetric with two c-bits being conveyed from Alice to Bob but only one from Bob to Alice.
We propose a protocol for the remote implementation of an arbitrary two-qubit unitary operation using pre-shared entanglement. In our protocol, Bob possesses an unknown two-qubit state, while Alice implements the required operations using calibrated black-box devices. Alice does not need to know the parameters of these operations, and Bob's unknown state never leaves his laboratory. The protocol is based on the Euler decomposition of single-qubit unitary operations and the Cartan decomposition of two-qubit unitary operations. First, we construct protocols for the remote implementation of the single-qubit operation U=Rz(γ)Rx(β)Rz(α)U = R_z(γ)R_x(β)R_z(α)U=Rz​(γ)Rx​(β)Rz​(α) using pre-shared Bell states. We then construct a protocol for the two-qubit operation Rzz(θ)R_{zz}(θ)Rzz​(θ) using a single pre-shared Bell state and extend it to Rxx(θ)R_{xx}(θ)Rxx​(θ) and Ryy(θ)R_{yy}(θ)Ryy​(θ) through local unitary transformations. These elementary protocols are finally combined to implement an arbitrary two-qubit unitary operation. A third party, Charlie, determines the required decomposition and provides Alice with the corresponding black-box devices before the protocol begins. Thus, neither Alice nor Bob requires the classical description of the unitary operation.
We present two protocols for the controlled remote implementation of quantum operations between three-party high-dimensional systems. Firstly, the controlled teleportation of an arbitrary unitary operation by bidirectional quantum state teleportaion (BQST) with high-dimensional systems is considered. Then, instead of using the BQST method, a protocol for controlled remote implementation of partially unknown operations belonging to some restricted sets in high-dimensional systems is proposed. It is shown that, in these protocols, if and only if the controller would like to help the sender with the remote operations, the controlled remote implementation of quantum operations for high-dimensional systems can be completed.
We present the generalized version of Wang's protocol[A.M.Wang, Phys.Rev.A 74,032317 (2006)] for the remote implementation(sometimes referred to as quantum remote control) of partially unknown quantum operations. The protocol only requires no more than half of the entanglements used in Bidirectional Quantum State Teleportation. We also propose a protocol for another form of quantum remote control. It can remotely implement a unitary operation which is a combination of the projective representations of a group. Moreover, we prove that the Schmidt rank of the entanglements cannot not be less than the number of controlled parameters of the operations, which for the first time gives a lower bound on entanglement costs in remote implementation of quantum operations.
Abstract In this work, we present two schemes for the bidirectional controlled remote implementation of quantum operations. Firstly, the bidirectional controlled teleportation of two arbitrary unitary operations is considered. Then, the scheme for bidirectional controlled remote implementation of partially unknown operations belong to some restricted sets is proposed. In these schemes, two users are not only senders but also receivers. It is shown that, in these schemes, only if the two senders and controller collaborate with each other, the bidirectional controlled remote implementation of quantum operations can be completed successfully.