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arXiv:quant-ph/0508054

Implementation of non-local operations for arbitrary high-dimensional systems with qubit quantum channel

Hao‐Sheng Zeng, Yong-Guang Shan, Jianjun Nie, Le‐Man Kuang

Published Aug 6, 2005• 0 citations
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Abstract

We propose a method to implement a kind of non-local operations between spatially separated two systems with arbitrary high-dimensions by using only low-dimensional qubit quantum channels and classical bit communications. The result may be generalized straightforwardly to apply for multiple systems, each of them with arbitrary dimensions. Compared with existed approaches, our method can economize classical resources and the needed low-dimensional quantum channels may be more easily established in practice. We also show the construction of the non-local quantum XOR gate for qutrit systems in terms of the obtained non-local operations as well as some single qutrit local gates.

DOI
10.48550/arxiv.quant-ph/0508054

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We propose a method to implement a kind of non-local operations between spatially separated two systems with arbitrary high-dimensions by using only low-dimensional qubit quantum channels and classical bit communications. The result may be generalized straightforwardly to apply for multiple systems, each of them with arbitrary dimensions. Compared with existed approaches, our method can economize classical resources and the needed low-dimensional quantum channels may be more easily established in practice. We also show the construction of the non-local quantum XOR gate for qutrit systems in terms of the obtained non-local operations as well as some single qutrit local gates.
Quantum computers are very efficient in terms of speed and security. Decoherence and architectural complexity restrict the control of sensitive quantum information as the number of qubits in a quantum computer increases. Therefore, it is more convenient to make a device with multiple quantum processors with small number of qubits instead of making a device with a large number of qubits quantum processors. The implementation of controlled unitary gates is a problem in such nonlocal systems. The methods in the literature for this problem use entangled qubit pairs, classical communication channels and classical bits. The existing methods perform some unitary operations on the target state for reconstruction after sending information through classical communication channels and applying quantum measurements on control states. In this study, a generalized method for nonlocal implementation of multi-qubit controlled unitary quantum gates with quantum channel is proposed. The proposed method can implement any controlled gate on the control and target qubits that are far from each other in terms of location. The method does not require classical channels and classical bits, any extra unitary operation for reconstruction. The proposed method is both more secure and uses less resources for operations than the other hybrid methods in the literature. Comparisons with existing studies are given in terms of required entangled qubit pairs, classical channels and bits, extra unitary operations for reconstruction the target state, and the advantages of the proposed method are revealed.
Consider two remote laboratories: Alice holds an unknown input |ξ⟩ and a private single-qubit unitary UA, while Bob holds |η⟩ and UB. We present an exact local-operations-and-classical-communication protocol that produces UB|ξ⟩ at Alice’s site and UA|η⟩ at Bob’s site only after the controller labels are released. A five-qubit channel first relocates the input states; operation delivery then uses a five-particle channel of local dimension d, a 2d-outcome nonsymmetric qubit–qudit measurement, and label-dependent inverse maps. We give the qutrit case explicitly and one correction formula for arbitrary d. A dual-layer construction lets Charlie authorize both directions, while David and Fiona authorize one direction each. In the ideal lossless model, all admissible branches have recovery maps and their probabilities sum to unity. The basic qutrit protocol uses ten entangled carriers and 13 fixed-length classical bits; the dual-layer protocol uses fourteen carriers and 19 bits. Thus, dimensional flexibility and direction-selective control are obtained at increased resource-protection and signaling costs. This is an exact theoretical remote-operation construction, not an experimentally mature or composably secure communication protocol.
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.
Instantaneous nonlocal quantum computation refers to a process in which spacelike separated parties simulate a nonlocal quantum operation on their joint systems through the consumption of pre-shared entanglement. To prevent a violation of causality, this simulation succeeds up to local errors that can only be corrected after the parties communicate classically with one another. However, this communication is non-interactive, and it involves just the broadcasting of local measurement outcomes. We refer to this operational paradigm as local operations and broadcast communication (LOBC) to distinguish it from the standard local operations and (interactive) classical communication (LOCC). In this paper, we show that an arbitrary two-qubit gate can be implemented by LOBC with ϵ-error using O(log(1/ϵ)) entangled bits (ebits). This offers an exponential improvement over the best known two-qubit protocols, whose ebit costs behave as O(1/ϵ). We also consider the family of binary controlled gates on dimensions dA⊗ dB. We find that any hermitian gate of this form can be implemented by LOBC using a single shared ebit. In sharp contrast, a lower bound of log dB ebits is shown in the case of generic (i.e. non-hermitian) gates from this family, even when dA= 2. This demonstrates an unbounded gap between the entanglement costs of LOCC and LOBC gate implementation. Whereas previous lower bounds on the entanglement cost for instantaneous nonlocal computation restrict the minimum dimension of the needed entanglement, we bound its entanglement entropy. To our knowledge this is the first such lower bound of its kind.
We present two optimal schemes for non-local implementing a single-qubit rotation operation via a maxi-mally entangled quantum channel. We report on the quantitative relations between the quantum action,entangled and classical communication resources required in the implementation. We also put forward twoschemes for conclusive implementing the non-local quantum single-qubit rotation via a partially entangledquantum channel. Both these methods can appropriately be referred to as qubit-assisted processes.