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arXiv:quant-ph/0601045

Entanglement and Teleportation of Gaussian States of the Radiation Field

Paulina Marian, Tudor A. Marian, Horia Scutaru

Published Jan 7, 2006• 0 citations
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Abstract

We propose a reliable entanglement measure for a two-mode squeezed thermal state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable states of the same kind. The requisite Uhlmann fidelity of a pair of two-mode squeezed thermal states is evaluated as the maximal transition probability between two four-mode purifications. By applying the Peres-Simon criterion of separability we find the closest separable state. This enables us to derive an insightful expression of the amount of entanglement. Then we apply this measure of entanglement to the study of the Braunstein-Kimble protocol of teleportation. We use as input state in teleportation a mixed one-mode Gaussian state. The entangled state shared by the sender (Alice) and the receiver (Bob) is taken to be a two-mode squeezed thermal state. We find that the properties of the teleported state depend on both the input state and the entanglement of the two-mode resource state. As a measure of the quality of the teleportation process, we employ the Uhlmann fidelity between the input and output mixed one-mode Gaussian states.

DOI
10.48550/arxiv.quant-ph/0601045

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We propose a reliable entanglement measure for a two-mode squeezed thermal state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable states of the same kind. The requisite Uhlmann fidelity of a pair of two-mode squeezed thermal states is evaluated as the maximal transition probability between two four-mode purifications. By applying the Peres-Simon criterion of separability we find the closest separable state. This enables us to derive an insightful expression of the amount of entanglement. Then we apply this measure of entanglement to the study of the Braunstein-Kimble protocol of teleportation. We use as input state in teleportation a mixed one-mode Gaussian state. The entangled state shared by the sender (Alice) and the receiver (Bob) is taken to be a two-mode squeezed thermal state. We find that the properties of the teleported state depend on both the input state and the entanglement of the two-mode resource state. As a measure of the quality of the teleportation process, we employ the Uhlmann fidelity between the input and output mixed one-mode Gaussian states.
We propose a reliable entanglement measure for a two-mode squeezed thermal state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable states of the same kind. The requisite fidelity of a pair of two-mode squeezed thermal states is exactly evaluated. By applying the Peres-Simon criterion of separability, we find the closest separable state. This enables us to establish an insightful expression of the amount of entanglement. We also derive the relative entropy of a two-mode squeezed thermal state with respect to another one. This quantity is exploited in the pure-state case by undertaking a similar approximation for the entropic entanglement of a two-mode squeezed vacuum state. Comparison with previous work lends support to our approach that replaces the reference set of all separable Gaussian states by its subset consisting only of the squeezed thermal ones.
We study the Braunstein-Kimble protocol for the continuous variable teleportation of a coherent state. We determine lower and upper bounds for the optimal fidelity of teleportation, maximized over all local Gaussian operations for a given entanglement of the two-mode Gaussian state shared by the sender (Alice) and the receiver (Bob). We also determine the optimal local transformations at Alice and Bob sites and the corresponding maximum fidelity when one restricts to local trace-preserving Gaussian completely positive maps.
Quantum teleportation is a fundamental protocol in quantum information science. It represents a critical resource for quantum communication and distributed quantum computing. We derive an analytical expression of the fidelity of teleportation of an input squeezed thermal state using for teleportation a bipartite Gaussian resource state shared between Alice and Bob. Each mode of the resource state is susceptible to the influence of the environment. We employ the characteristic function approach in conjunction with the covariance matrix formalism. The fidelity of teleportation is expressed in terms of input and resource state covariance matrices. We investigate, as an example, the feasibility of secure quantum teleportation of a squeezed thermal state using a two-mode resource state whose modes are placed in separate thermal baths. A successful quantum teleportation requires meeting two criteria: the presence of two-way quantum steering and a teleportation fidelity exceeding the classical threshold. The quantum steering is by nature asymmetric and has found applications in quantum cryptography and secure quantum teleportation. Weak squeezing and a high number of average thermal photons in the input states lead to an increase in the fidelity of teleportation. Generally, steering disappears much faster than the fidelity of teleportation decreases below its classical limit.
In contrast to discrete-variable teleportation, a quantum state is imperfectly transferred from a sender to a remote receiver in a continuous-variable setting. We recall the ingenious scheme proposed by Braunstein and Kimble for teleporting a one-mode state of the quantum radiation field. By analyzing this protocol, we have previously proven the factorization of the characteristic function of the output state. This indicates that teleportation is a noisy process that alters, to some extent, the input state. Teleportation with a two-mode Gaussian EPR state can be described in terms of the superposition of a distorting field with the input one. Here we analyze the one-mode Gaussian distorting-field state. Some of its most important properties are determined by the statistics of a positive EPR operator in the two-mode Gaussian resource state. We finally examine the fidelity of teleportation of a coherent state when using an arbitrary resource state.
The entanglement properties of non-Gaussian states are investigated, which are obtained by performing the photon addition, photon subtraction, photon-addition-then-subtraction, and photon-subtraction-then-addition operations on the two-mode squeezed vacuum state. We show that the partial von Neumann entropy of all the resulting states is greater than that of the original squeezed state, but only the photon-subtracted states and the photon-added-then-subtracted states have the stronger Einstein-Podolsky-Rosen correlation than the original squeezed state. Quantum teleportation of Braunstein and Kimble protocol is studied for coherent states, squeezed states, and mixed Gaussian states with the non-Gaussian entangled resources. For all the states to be teleported, the fidelity with the photon-subtracted and the photon-added-then-subtracted entangled resources is higher than that with the two-mode squeezed vacuum resource. Based on Bures fidelity, we find that quantum teleportation for mixed and classical single-mode Gaussian states is more faithful than for single-mode Gaussian states with high purity and nonclassicality.
The analytical formula for the fidelity of teleportation from Alice to Bob of a thermal state used as an input state is derived using the characteristic function approach. As a resource state it is used a continuous-variable two-mode bosonic state shared between Alice and Bob and the system is placed in a contact with a general environment. The quantum fidelity depends on the average thermal photon number of the input sate and on the parameters of the resource state and environment. In order to evaluate the parameters required for a successful quantum teleportation of a thermal state we estimate the evolution of the entanglement and fidelity of teleportation in the case when Alice and Bob share an initial squeezed thermal state in contact with a thermal bath. We work in the framework of the theory of open quantum systems based on completely positive dynamical semigroups and we use the Gorini-Kossakowski-Lindblad-Sudarshan master equation in order to describe the evolution of the system. The conditions for a successful quantum teleportation are the fidelity of teleportation larger than the threshold value of the classical fidelity of teleportation and entanglement present in the resource state. In order to assure the fulfillment of these conditions it is necessary to select the time of teleportation and to adjust the parameters characterizing the system (squeezing between the modes of the resource state, their average numbers of thermal photons and frequencies), and the temperature of the thermal reservoir. It is shown that in similar physical conditions, the fidelity of teleportation of the thermal states is larger than that one for a coherent state.