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Teleportation of general finite dimensional quantum systems

Sergio A Albeverio, Shao-Ming Fei, Fakultät für Mathematik, Ruhr-universität Bochum, D- Bochum

Published Jan 1, 2000• 42 citations
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Abstract

Teleportation of finite dimensional quantum states by a non-local entangled state is studied. For a generally given entangled state, an explicit equation that governs the teleportation is presented. Detailed examples and the roles played by the dimensions of the Hilbert spaces related to the sender, receiver and the auxiliary space are discussed.

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Teleportation of finite dimensional quantum states by a non-local entangled state is studied. For a generally given entangled state, an explicit equation that governs the teleportation is presented. Detailed examples and the roles played by the dimensions of the Hilbert spaces related to the sender, receiver and the auxiliary space are discussed.
Teleportation for pure states, mixed states with standard and optimal protocols are introduced and investigated systematically. An explicit equation governing the teleportation of finite dimensional quantum pure states by a generally given non-local entangled state is presented. For the teleportation of a mixed state with an arbitrary mixed state resource, an explicit expression is obtained for the quantum channel associated with the standard teleportation protocol. The corresponding transmission fidelity is calculated. It is shown that the standard teleportation protocol is not optimal. The optimal quantum teleportation is further studied, its fidelity is given and shown to be related to the fully entangled fraction of the quantum resource, rather than the single fraction as in the standard teleportation protocol.
Quantum teleportation is the disembodied transport of an unknown quantum state |ψ〉 from one place to another. All protocols for accomplishing such transport require nonlocal correlations, or entanglement, between systems shared by sender and receiver. The sender is normally called Alice and the receiver is called Bob . Most attention has focused on teleporting the states of finite-dimensional systems, such as the two-dimensional polarization of a photon or the discrete level structure of an atom. First proposed in 1993 by Charles Bennett and his colleagues [17, 24, 138] quantum teleporation thus allows physicists to take a photon or any other quantum scale particle such as an atom and transfer its properties (such as the polarization) to another photon even if the two photons are on opposite sides of the galaxy. This scheme transports the particle’s properties to the remote location and not the particle itself. The state of the original particle must be destroyed to create an exact reconstruction at the other end. This is a consequence of the no cloning theorem. A role in the teleportation scheme is played by an entangled ancillary pair of particles which will be initially shared by Alice and Bob.
We present a general solution to the construction of such an entanglement in an N-dimensional Hilbert space that effects state teleportation. For N>2N>2N>2 the construction is nonunique and can be chosen to transfer the state to a preselected basis. We discuss possible applications to state preparation in various physical systems.
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.
Through a detailed study of nondestructive controlled quantum teleportation (NCQT) with two outputs in three-dimensional and high-dimensional Hilbert spaces, we propose an NCQT scheme that synchronously and probabilistically teleports N arbitrary unknown d-dimensional single-particle states from one sender to N different receivers under the supervision of a controller, by using a d-dimensional partially entangled (2N+1)-particle state as the quantum channel. The protocol succeeds if and only if all receivers recover their respective target states, and the optimal success probability of the N-output NCQT is determined by the minimum superposition coefficient of each product two-qudit state in the partially entangled channel. In our scheme, each receiver introduces an auxiliary qubit to assist in the local recovery test. When the auxiliary-qubit measurement outcome is |0⟩, the receiver can restore the target state; when the outcome is |1⟩, the corresponding unknown original state is retained by the sender. Accordingly, the N-output controlled teleportation process can be repeated as many times as additional quantum channels are available after a failed attempt. The results show that weakly entangled channels can still realize N-output controlled teleportation through sufficiently many repetitions, whereas strongly entangled channels require only a small number of repetitions to achieve the same goal.
Quantum teleportation has been introduced by Benett et al. and dis cussed by a number of authors in the framework of the singlet state. Recentl y, a rigorous formulation of the teleportation problem of arbitrary quantum states by means of quantum channel was given in [IOS] based on the general c hannel theoretical formulation of the quantum gates introduced in [OW]. In t his note we discuss a generalization of the scheme proposed in [IOS] and we give a general method to solve the teleportation problem in spaces of arbitr ary finite dimensions.