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Physical Review A

Quantum teleportation of composite systems via mixed entangled states

Somshubhro Bandyopadhyay, Barry C. Sanders

Published Sep 12, 2006• 31 citations
Publisher

Abstract

We analyze quantum teleportation for composite systems, specifically for concatenated teleporation (decomposing a large composite state into smaller states of dimension commensurate with the channel) and partial teleportation (teleporting one component of a larger quantum state). We obtain an exact expression for teleportation fidelity that depends solely on the dimension and singlet fraction for the entanglement channel and entanglement (measures by $I$ concurrence) for the state; in fact quantum teleportation for composite systems provides an operational interpretation for $I$ concurrence. In addition we obtain tight bounds on teleportation fidelity and prove that the average fidelity approaches the lower bound of teleportation fidelity in the high-dimension limit.

DOI
10.1103/physreva.74.032310

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We analyze quantum teleportation for composite systems, specifically for concatenated teleporation (decomposing a large composite state into smaller states of dimension commensurate with the channel) and partial teleportation (teleporting one component of a larger quantum state). We obtain an exact expression for teleportation fidelity that depends solely on the dimension and singlet fraction for the entanglement channel and entanglement (measures by III concurrence) for the state; in fact quantum teleportation for composite systems provides an operational interpretation for III concurrence. In addition we obtain tight bounds on teleportation fidelity and prove that the average fidelity approaches the lower bound of teleportation fidelity in the high-dimension limit.
We analyse the problem of transmitting a number of unknown quantum states or one composite system in one go. We derive a lower bound on the performance of such process, measured in the entanglement fidelity. The obtained bound is effectively computable and outperforms the explicit values of the entanglement fidelity calculated for the pre-existing variants of the port-based protocols, allowing for teleportation of a much larger amount of quantum information. The comparison with the exact formulas and similar analysis for the probabilistic scheme is also discussed. In particular, we present the closed-form expressions for the entanglement fidelity and for the probability of success in the probabilistic scheme in the qubit case in the picture of the spin angular momentum.
Teleporation with partially entangled quantum channel cannot achieve unit fidelity and unit probability. We show that the conditions for faithful teleportation of a pure state or a mixed state can be described by the same general relation ρqρm=pI\rho_{q}\rho_{m} = pIρq​ρm​=pI, where ρq\rho_{q}ρq​ is the reduced density matrix of the quantum channel, ρm\rho_{m}ρm​ is the reduced density matrix of the measurement basis, and ppp is the probability of faithful teleportaion. These conditions are unchanged under unitary transformation. We also prove that using a four dimensional partially entangled quantum channel, it is not possible to find a set of orthogonal measurement basis that contains more than two faithfully teleporting measurements.
Quantum teleportation efficiently transfers quantum information between distant locations by utilizing a pre-established composite system. Assessing the effectiveness of teleportation hinges on its fidelity, representing the similarity between input and output states. This fidelity, in turn, relies on a singlet fraction, quantifying the resemblance of the composite system to maximally entangled states. The relation between teleportation fidelity and singlet fraction given by Horodecki et al. [Phys. Rev. A 60, 1888 (1999)] does not hold for distinguishable particles with multiple degrees of freedom or indistinguishable particles with single or multiple degrees of freedom. In this paper, we propose generalized expressions for teleportation fidelity and singlet fraction and derive their relations, applicable to both distinguishable and indistinguishable particles with single or multiple degrees of freedom. We derive an upper bound for the generalized singlet fraction for distinguishable particles using the monogamy of singlet fraction by Kay et al. [Phys. Rev. Lett. 103, 050501 (2009)]. We also show how our relation helps characterize different types of composite states in terms of their distinguishability, separability, the presence of maximally entangled structure, and the number of degrees of freedom. We complement our theory with two practical illustrations. First, we demonstrate two counter-intuitive values of generalized singlet fraction using our optical circuit and the circuit of Li et al. [Phys. Rev. Lett. 120, 050404 (2018)]. Finally, we show that using an additional degree of freedom as an ancilla instead of a particle can be advantageous in quantum cryptographic protocols.
In this article, we find a theorem that gives a relation between the maximal fidelity of teleportation and the concurrence of the inseparable XXX state used as a quantum channel in this process. Furthermore, we evaluate the concurrence of the output states generated by the local and nonlocal asymmetric broadcasting of entanglement and prove that the concurrence is greater in the case of nonlocal broadcasting. We analyze the possibility of using the output states obtained by the broadcasting of entanglement as quantum channels in quantum teleportation. We prove, with the help of the above-mentioned theorem, that all the inseparable states given by the local and nonlocal asymmetric broadcasting of entanglement are useful for quantum teleportation. Finally, we show that the maximal fidelity of teleportation is greater in the case when the second scenario is used, i.e., when the quantum channel is generated by the nonlocal asymmetric broadcasting of entanglement.
We investigate quantum teleportation through dissipative channels and calculate teleportation fidelity as a function of damping rates. It is found that the average fidelity of teleportation and the range of states to be teleported depend on the type and rate of the damping in the channel. Using the fully entangled fraction, we derive two bounds on the damping rates of the channels: one is to beat the classical limit and the second is to guarantee the nonexistence of any other copy with better fidelity. The effect of the initially distributed maximally entangled state on the process is presented; the concurrence and the fully entangled fraction of the shared states are discussed. We intend to show that prior information on the dissipative channel and the range of qubit states to be teleported is helpful for the evaluation of the success of teleportation, where success is defined as surpassing the fidelity limit imposed by the fidelity of the 1-to-2 optimal cloning machine for the specific range of qubits.