We investigate remote state preparation (RSP) via a non-maximally entangled channel for three cases: a general qubit; a special ensemble of qubits (qubit states on the equator of the Bloch sphere); and an asymptotic limit of N copies of a general state. The results show that the classical communication cost of RSP for the two latter cases can be less than that of teleportation, but for the first case, in a restricted setting, the classical communication cost is equal to that of teleportation. Whether or not this is the case for a more general setting is still an open question.
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.
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.
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 study the following task: Alice is given a classical description of a rank-
k projector
P on
Cd, and Alice and Bob want to prepare the quantum state
P/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/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) bits of communication, can distill
logd ebits of entanglement, and conversely, any state that can distill
logd 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/k, and give a new entanglement-assisted communication protocol for the equality function that uses
21logn+O(1) many ebits, and
O(1) communication.
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.