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arXiv:quant-ph/0205009 · Physical Review A

Remote state preparation without oblivious conditions

A. Hayashi, T. Hashimoto, M. Horibe

Published May 13, 2003• 120 citations
arXivHTMLPDF

Abstract

In quantum teleportation, neither Alice nor Bob acquires any classical knowledge on teleported states. The teleportation protocol is said to be oblivious to both parties. In remote state preparation (RSP), it is assumed that Alice is given complete classical knowledge on the state that is to be prepared by Bob. Recently, Leung and Shor [e-print quant-ph/0201008] showed that the same amount of classical information as that in teleportation needs to be transmitted in any exact and deterministic RSP protocol that is oblivious to Bob. Assuming that the dimension of subsystems in the prior-entangled state is the same as the dimension of the input space, we study similar RSP protocols, but not necessarily oblivious to Bob. We show that in this case Bob's quantum operation can be safely assumed to be a unitary transformation. We then derive an equation that is a necessary and sufficient condition for such a protocol to exist. By studying this equation, we show that one-qubit RSP requires two classical bits of communication, which is the same amount as in teleportation, even if the protocol is not assumed oblivious to Bob. For higher dimensions, it is still an open question whether the amount of classical communication can be reduced by abandoning oblivious conditions.

DOI
10.1103/physreva.67.052302

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In quantum teleportation, neither Alice nor Bob acquires any classical knowledge on teleported states. The teleportation protocol is said to be oblivious to both parties. In remote state preparation (RSP), it is assumed that Alice is given complete classical knowledge on the state that is to be prepared by Bob. Recently, Leung and Shor [e-print quant-ph/0201008] showed that the same amount of classical information as that in teleportation needs to be transmitted in any exact and deterministic RSP protocol that is oblivious to Bob. Assuming that the dimension of subsystems in the prior-entangled state is the same as the dimension of the input space, we study similar RSP protocols, but not necessarily oblivious to Bob. We show that in this case Bob's quantum operation can be safely assumed to be a unitary transformation. We then derive an equation that is a necessary and sufficient condition for such a protocol to exist. By studying this equation, we show that one-qubit RSP requires two classical bits of communication, which is the same amount as in teleportation, even if the protocol is not assumed oblivious to Bob. For higher dimensions, it is still an open question whether the amount of classical communication can be reduced by abandoning oblivious conditions.
Quantum teleportation uses prior entanglement and forward classical communication to transmit one instance of an unknown quantum state. Remote state preparation (RSP) has the same goal, but the sender knows classically what state is to be transmitted. We show that the asymptotic classical communication cost of RSP is one bit per qubit--half that of teleportation--and even less when transmitting part of a known entangled state. We explore the tradeoff between entanglement and classical communication required for RSP, and discuss RSP capacities of general quantum channels.
A necessary and sufficient condition is given for general exact remote state preparation (RSP) protocols to be oblivious, that is, no information about the target state can be retrieved from the classical message. A criterion in terms of commutation relations is also derived for the existence of deterministic exact protocols in which Alice's measurement eigenstates are related to each other by fixed linear operators similar to Bob's unitaries. For nonmaximally entangled resources, it provides an easy way to search for RSP protocols. As an example, we show how to reduce the case of partially entangled resources to that of maximally entangled ones, and we construct RSP protocols exploiting the structure of the irreducible representations of Abelian groups.
Quantum teleportation uses prior shared entanglement and classical communication to send an unknown quantum state from one party to another. Remote state preparation (RSP) is a similar distributed task in which the sender knows the entire classical description of the state to be sent. (This may also be viewed as the task of nonoblivious compression of a single sample from an ensemble of quantum states.) We study the communication complexity of approximate RSP (ARSP) in which the goal is to prepare an approximation of the desired quantum state. Jain (Quant. Inf. & Comp., 2006) showed that the worst-case communication complexity of ARSP can be bounded from above in terms of the maximum possible information in an encoding. He also showed that this quantity is a lower bound for communication complexity of (exact) remote state preparation. In this paper, we tightly characterize the worst-case and average-case communication complexity of remote state preparation in terms of nonasymptotic information-theoretic quantities. We also show that the average-case communication complexity of RSP can be much smaller than the worst-case one. In the process, we show that nnn bits cannot be communicated with less than nnn transmitted bits in local operations and classical communication protocols. This strengthens a result due to Nayak and Salzman (J. ACM, 2006) and may be of independent interest.
We characterize the class of remote state preparation (RSP) protocols that use only forward classical communication and entanglement, deterministically prepare an exact copy of a general state, and do so obliviously-without leaking further information about the state to the receiver. We prove that any such protocol can be modified to require from the sender only a single specimen of the state, without increasing the classical communication cost. This implies Lo's conjectured lower bound on the cost for these protocols. We relate our RSP protocols to the private quantum channels and establish a one-to-one correspondence between them.
Superdense coding and quantum teleportation are two phenomena which were not possible without prior entanglement. In superdense coding, one sends n bits of information using n/2 qubits in the presence of shared entanglement. However, we show that n bits of information cannot be sent with less than n bits of communication in LOCC protocols even in the presence of prior entanglement. This is an interesting result which will be used in the rest of this thesis. \n \nQuantum teleportation uses prior entanglement and classical communication to send \nan unknown quantum state. Remote state preparation (RSP) is the same distributed task, \nbut in the case that the sender knows the description of the state to be sent, completely. \nWe study the communication complexity of approximate remote state preparation in which \nthe goal is to prepare an approximation of the desired quantum state. Jain showed that the worst-case error communication complexity of RSP can be bounded from above in terms of the maximum possible information in an encoding [18]. He also showed that this quantity is a lower bound for communication complexity of exact remote state preparation [18]. \nIn this thesis, we characterize the worst-case error and average-case error communication complexity of remote state preparation in terms of non-asymptotic information-theoretic quantities. We also utilize the bound we derived for the communication complexity of LOCC protocols in the first part of the thesis, to show that the average-case error communication complexity of RSP can be much smaller than the worst-case.
We introduce a multipartite communication scheme, with the aim to enable the senders to remotely and obliviously provide the receivers with an arbitrary amount of multipartite entanglement. The scheme is similar to remote state preparation (RSP). However, we show that even though the receivers are restricted to local unitary operations, the required resources for remote entanglement preparation are less than for RSP. In order to do so we introduce a canonical form of arbitrary multipartite pure states describing an arbitrary number of qubits. Moreover, we show that if the receivers are enabled to perform arbitrary local operations and classical communication, the required resources can drastically be reduced. We employ this protocol to derive robust entanglement purification protocols for arbitrary pure states and show that it can also be used for sending classical information.