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arXiv:0705.2162 · Physical Review A

Separability of entangled qutrits in noisy channels

Agata Chęcińska, Krzysztof Wódkiewicz

Published Nov 9, 2007• 34 citations
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Abstract

We present an analysis of noisy atomic channels involving qutrits. We choose a three-level atom with $\mathsf{V}$ configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (eight-dimensional) are used to describe the qutrit density operator. We introduce quantum quasidistributions for qutrits that provide a simple description of entanglement. Studying the time evolution for the atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition in the presence of spontaneous emission noise. The influence of spontaneous emission on the separability of Werner states for qutrit and qubit states is compared.

DOI
10.1103/physreva.76.052306

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We present an analysis of noisy atomic channels involving qutrits. We choose a three-level atom with V\mathsf{V}V configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (eight-dimensional) are used to describe the qutrit density operator. We introduce quantum quasidistributions for qutrits that provide a simple description of entanglement. Studying the time evolution for the atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition in the presence of spontaneous emission noise. The influence of spontaneous emission on the separability of Werner states for qutrit and qubit states is compared.
We present an analysis of spontaneous emission in a 3-level atom as an example of a qutrit state under the action of noisy quantum channels. We choose a 3-level atom with V-configuration to be the qutrit state. Gell-Mann matrices and a generalized Bloch vector (8-dimensional) are used to describe the qutrit density operator. Using the time-evolution equations of atomic variables we find the Kraus representation of spontaneous emission quantum channel (SE channel). Furthermore, we consider a generalized Werner state of two qutrits and investigate the separability condition. We give similar analysis of spontaneous emission for qubit channels. The influence of spontaneous emission on the separability of Werner states for qutrit and qubit states is compared.
Two important classes of quantum channels, namely, the Werner-Holevo and the Landau-Streater channels, are known to be related only in three dimensions, i.e., when acting on qutrits. In this work, the definition of the Landau-Streater channel is extended in such a way that it retains its equivalence to the Werner-Holevo channel in all dimensions. This channel is then modified to be representable as a model of noise acting on qudits. We then investigate properties of the resulting noisy channel and determine the conditions under which it cannot be the result of a Markovian evolution. Furthermore, we investigate its different capacities for transmitting classical and quantum information with or without entanglement. In particular, while the pure (or high-noise) Landau-Streater or the Werner-Holevo channel is entanglement breaking and hence has zero capacity, by finding a lower bound for the quantum capacity, we show that when the level of noise is lower than a critical value the quantum capacity will be nonzero. Surprisingly, this value turns out to be approximately equal to 0.4 in all dimensions. Finally, we show that, in even dimensions, this channel has a decomposition in terms of unitary operations. This is in contrast with the three-dimensional, case where it has been proved that such a decomposition is impossible, even in terms of other quantum maps.
Two important classes of quantum channels, namly the Werner-Holevo and the Landau-Streater channels are known to be related only in three dimensions, i.e. when acting on qutrits. In this work, definition of the Landau-Streater channel is extended in such a way which retains its equivalence to the Werner-Holevo channel in all dimensions. This channel is then modified to be representable as a model of noise acting on qudits. We then investigate propeties of the resulting noisy channel and determine the conditions under which it cannot be the result of a Markovian evolution. Furthermore, we investigate its different capacities for transmitting classical and quantum information with or without entanglement. In particular, while the pure (or high noise) Landau-Streater or the Werner-Holevo channel is entanglement breaking and hence has zero capacity, by finding a lower bound for the quantum capacity, we show that when the level of noise is lower than a critical value the quantum capacity will be non-zero. Surprizingly this value turns out to be approximately equal to 0.40.40.4 in all dimensions. Finally we show that, in even dimension, this channel has a decomposition in terms of unitary operations. This is in contrast with the three dimensional case where it has been proved that such a decomposition is impossible, even in terms of other quantum maps.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
Abstract Quantum teleportation is the fundamental communication unit in quantum communication. Here, a three-level system is selected for storing and transmitting quantum information, due to its unique advantages, such as lower cost than a higher-level system and higher capacity and security than a two-level system. It is known that the key procedure for perfect teleportation is the distribution of entanglement through quantum channel. However, amounts of noise existing in the quantum channel may interfere the entangled state, causing the degradation of quantum entanglement. In the physical implementations of quantum communication schemes, noise acting on the carriers of successive transmissions often exhibits some correlations, which is the so called quantum memory channel. In this paper, a memory channel model during the entanglement distribution phase is constructed and the uniform expression of the evolution of a two-qutrit entangled state under different kinds of correlated noise is derived. Finally, Pauli noise and amplitude damping noise as the typical noise source are considered to analyze the influence of memory effects of noise on qutrit teleportation. It is expected to show that three-level teleportation under these two types of channels can generally enhance the robustness to noise if the Markovian correlations of quantum channel are taken into consideration.