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arXiv:1405.5309

Optimizing classical communication in remote preparation of a general pure qubit

Congyi Hua, Yixin Chen

Published May 21, 2014• 0 citations
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Abstract

How to uses shared entanglement and forward classical communication to remotely prepare an arbitrary (mixed or pure) state has been fascinating quantum information scientists. A constructive scheme has been given by Berry for remotely preparing a general pure state with a pure entangled state and finite classical communication. Based on this scheme, for high-dimensional systems it is possible to use a coding of the target state to optimize the classical communication cost. Unfortunately, for low-dimensional systems such as a pure qubit the coding method is inapplicable. Because qubit plays a central role in quantum information theory, we propose an optimization procedure which can be used to minimize the classical communication cost in the remote preparation of a general pure qubit. Interestingly, our optimization procedure is linked to the uniform arrangement of $N$ points on the Bloch sphere, which provides a geometric description.

DOI
10.48550/arxiv.1405.5309

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How to uses shared entanglement and forward classical communication to remotely prepare an arbitrary (mixed or pure) state has been fascinating quantum information scientists. A constructive scheme has been given by Berry for remotely preparing a general pure state with a pure entangled state and finite classical communication. Based on this scheme, for high-dimensional systems it is possible to use a coding of the target state to optimize the classical communication cost. Unfortunately, for low-dimensional systems such as a pure qubit the coding method is inapplicable. Because qubit plays a central role in quantum information theory, we propose an optimization procedure which can be used to minimize the classical communication cost in the remote preparation of a general pure qubit. Interestingly, our optimization procedure is linked to the uniform arrangement of NNN points on the Bloch sphere, which provides a geometric description.
Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a tradeoff between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this "universal" formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact tradeoff curve for memoryless sources of pure states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical tradeoff for visible quantum data compression. A variation of that method allows us to solve the even more general problem of preparing entangled states between sender and receiver (i.e., purifications of mixed state ensembles). The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding.
We study the following task: Alice is given a classical description of a rank-kkk projector PPP on Cd\mathbb{C}^dCd, and Alice and Bob want to prepare the quantum state P/kP/kP/k on Bob's side using shared entanglement and classical communication. The general form of this task is known as remote state preparation (RSP). We give nearly-matching lower and upper bounds for the entanglement cost and communication cost for RSP of the states P/kP/kP/k. Ours are the first nearly matching upper and lower bounds for RSP of mixed states, and in the special case of pure states, our lower bound outperforms the best previously known lower bound. Our results show that any pure entangled state that can be used to do RSP of these states with o(d)o(d)o(d) bits of communication, can distill log⁡d\log dlogd ebits of entanglement, and conversely, any state that can distill log⁡d\log dlogd ebits of entanglement can be used to do RSP of these states efficiently. As applications of our results, we rederive a previously-known incompressibility result for states of the form P/kP/kP/k, and give a new entanglement-assisted communication protocol for the equality function that uses 12log⁡n+O(1)\frac{1}{2}\log n + O(1)21​logn+O(1) many ebits, and O(1)O(1)O(1) communication.
Classical communication plays an important role in quantum information processing such as remote state preparation and quantum teleportation. First, in this paper, we present some simple faithful remote state preparation of an arbitrary n-qubit state by constructing entanglement resources and special measurement basis for the sender. Then to weigh the classical resource required, we present an information-theoretical model to evaluate the classical communication cost. By optimizing the classical communication in quantum protocols, we obtain the optimal classical communication cost. This model can also be applied to the quantum teleportation. Moreover, based on the present computation model, we reinvestigate some remote state preparation and teleportation protocols in which the classical communication cost was imperfectly computed. Finally, some problems will be presented.
Abstract—Remote state preparation is the variant of quantum state teleportation in which the sender knows the quantum state to be communicated. The original paper introducing teleportation established minimal requirements for classical communication and entanglement but the corresponding limits for remote state preparation have remained unknown until now: previous work has shown, however, that it not only requires less classical communication but also gives rise to a tradeoff between these two resources in the appropriate setting. We discuss this problem from first principles, including the various choices one may follow in the definitions of the actual resources. Our main result is a general method of remote state preparation for arbitrary states of many qubits, at a cost of 1 bit of classical communication and 1 bit of entanglement per qubit sent. In this “universal ” formulation, these ebit and cbit requirements are shown to be simultaneously optimal by exhibiting a dichotomy. Our protocol then yields the exact tradeoff curve for memoryless sources of pure states (including the case of incomplete knowledge of the ensemble probabilities), based on the recently established quantum-classical tradeoff for visible quantum data compression. A variation of that method allows us to solve the even more general problem of preparing entangled states between sender and receiver (i.e., purifications of mixed state ensembles). The paper includes an extensive discussion of our results, including the impact of the choice of model on the resources, the topic of obliviousness, and an application to private quantum channels and quantum data hiding. Index Terms—Cryptography, entanglement, large deviations, teleportation, tradeoff. I.
We study the amount of classical communication needed for distributed quantum-information processing. In particular, we introduce the concept of ``remote preparation'' of a quantum state. Given an ensemble of states, Alice's task is to help Bob in a distant laboratory to prepare a state of her choice. We find several examples of an ensemble with an entropy S where the remote preparation can be done with a communication cost lower than the amount (2S)(2S)(2S) required by standard teleportation. We conjecture that, for an arbitrary N-dimensional pure state, its remote preparation requires 2log2N2{\mathrm{log}}_{2}N2log2​N bits of classical communication, as in standard teleportation.
We experimentally demonstrate a protocol of probabilistic remote state preparation by virtue of entanglement, local operation and forward classical communications. Arbitrary polarization qubit state can be remotely prepared with this protocol. For both pure states and mixed states, the average efficiency is no less than 50% and the classical information cost is 1 bit. All 18 remotely prepared states were estimated with quantum state tomography system. The fidelities are all above 0.99, with average fidelity being 0.9956.