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arXiv:quant-ph/0308020 · Fortschritte der Physik

Teleportation with Partially Entangled States

Z. Kurucz, Mátyás Koniorczyk, József Janszky

Published Oct 1, 2001• 24 citations
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Abstract

The relations of antilinear maps, bipartite states and quantum channels is summarized. Antilinear maps are applied to describe bipartite states and entanglement. Teleportation is treated in this general formalism with an emphasis on conditional schemes applying partially entangled pure states. It is shown that in such schemes the entangled state shared by the parties, and those measured by the sender should “match” each other.

DOI
10.1002/1521-3978(200110)49:10/11<1019::aid-prop1019>3.0.co;2-z

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The relations of antilinear maps, bipartite states and quantum channels is summarized. Antilinear maps are applied to describe bipartite states and entanglement. Teleportation is treated in this general formalism with an emphasis on conditional schemes applying partially entangled pure states. It is shown that in such schemes the entangled state shared by the parties, and those measured by the sender should “match” each other.
The antilinear operator representation of bipartite pure states of the relative state formulation of quantum mechanics is applied to describe quantum teleportation schemes utilizing an arbitrary pure state as the entangled resource. Bennett type teleportation schemes with nonmaximally entangled pure states are characterized and the notion of 'entanglement matching' is introduced in general. Examples, including a scheme based on coherent-state superposition states of the electromagnetic field, are provided.
Quantum teleportation provides a way to transfer unknown quantum states from one system to another via an entangled state as a quantum channel without physical transmission of the object itself. The entangled channel, measurement performed by the sender (Alice), and classical information sent to the receiver (Bob) are three key ingredients in the procedure, which need to cooperate with each other. To study the relationship among the three parts, we propose a scheme for perfect teleportation of a qubit through a high-dimensional quantum channel in a pure state with two equal largest Schmidt coefficients. The scheme requires less entanglement of Alice's measurement but more classical bits than the original scheme via a Bell state. The two quantities increase with the entanglement of the quantum channel when its dimension is fixed and thereby can be regard as Alice's necessary capabilities to use the quantum channel. And the two capabilities appear complementary to each other.
A scheme for teleporting a two-particle entangled state via special partial two-particle entangled W state and partial two-particle entangled state is proposed.There is a certain probability of successful teleportion of a two-particle entangled state if the sender performs two Bell-state measurements while the receiver first makes a measurement in the basis of {|0〉,|1〉},then performs a controlled-not operation and furthur introduces an ancillary qubit and performs an appropriate unitary transformation at last.Analysis shows: when the quantum channel is of maximal entanglement,namely the channel is composed of a special two-particle entangled W state and a Bell state,the teleportion probability of this scheme reachs 2/3 and the teleportion effect of this scheme is between the scheme whose quantum channel is composed of W states and the one whose quantum channel is composed of Bell states.
Two-qubit quantum teleportation is closely related to quantum computation,so a teleportation protocol in which an arbitrary bipartite quantum state is perfectly teleported probabilistically from sender to receiver is proposed. One of 16 generalized non-maximally entangled Bell states (G states for simplicity) functions as quantum channel. The teleportation can be successfully realized with a certain probability if sender performs generalized Bell state measurements (G measurements) and receiver introduces an auxiliary particle and operates appropriate unitary transformations and single-qubit measurements. The probability of successful teleportation is determined by the smallest one among the coefficients' absolute values of the quantum channel.
It deserves mentioning that the quantum circuit, i.e. quantum logic network, is essential to the practical realization of teleportation in experiment. Using only one partially entangled pair, we first propose two novel strategies for probabilistically teleporting any partially entangled state of a bipartite system. The feature of the present protocol is to weaken the requirement for the quantum channel, and also to cut down the number of entangled particles initially shared by the sender and receiver. On the other hand, we explicitly construct the generalized measurement described by the positive operator-valued measure (POVM). Two kinds of efficient quantum circuits for implementing the teleportation are offered. In addition, we generalize the two-particle probabilistic teleportation to the system of many particles.