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arXiv:quant-ph/0508106 · Physical Review A

Concurrence versus purity: Influence of local channels on Bell states of two qubits

Mário Ziman, Vladimír Bužek

Published Nov 22, 2005• 57 citations
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Abstract

We analyze how a maximally entangled state of two qubits (e.g., the singlet ${\ensuremath{\psi}}_{s}$) is affected by the action of local channels described by completely positive maps $\mathcal{E}$. We analyze the concurrence and the purity of states ${\ensuremath{\varrho}}_{\mathcal{E}}=\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}[{\ensuremath{\psi}}_{s}]$. Using the concurrence-versus-purity phase diagram we characterize local channels $\mathcal{E}$ by their action on the singlet state ${\ensuremath{\psi}}_{s}$. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.

DOI
10.1103/physreva.72.052325

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Quantum Information and Computation•May 1, 2013•1 citation
We analyze how a maximally entangled state of two qubits (e.g., the singlet \ensuremathψs{\ensuremath{\psi}}_{s}\ensuremathψs​) is affected by the action of local channels described by completely positive maps E\mathcal{E}E. We analyze the concurrence and the purity of states \ensuremathϱE=E\ensuremath⨂I[\ensuremathψs]{\ensuremath{\varrho}}_{\mathcal{E}}=\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}[{\ensuremath{\psi}}_{s}]\ensuremathϱE​=E\ensuremath⨂I[\ensuremathψs​]. Using the concurrence-versus-purity phase diagram we characterize local channels E\mathcal{E}E by their action on the singlet state \ensuremathψs{\ensuremath{\psi}}_{s}\ensuremathψs​. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.
We analyze how a maximally entangled state of two qubits (e.g., the singlet {psi}{sub s}) is affected by the action of local channels described by completely positive maps E. We analyze the concurrence and the purity of states {rho}{sub E}=ExI[{psi}{sub s}]. Using the concurrence-versus-purity phase diagram we characterize local channels E by their action on the singlet state {psi}{sub s}. We specify a region of the concurrence-versus-purity diagram that is achievable from the singlet state via the action of unital channels. We show that even the most general (including nonunital) local channels acting just on a single qubit of the original singlet state cannot generate the maximally entangled mixed states. We study in detail various time evolutions of the original singlet state induced by local Markovian semigroups. We show that the decoherence process is represented in the concurrence-versus-purity diagram by a line that forms the lower bound of the achievable region for unital maps. On the other hand, the depolarization process is represented by a line that forms the upper bound of the region of maps induced by unital maps.
We analyze how entanglement between two components of a bipartite system behaves under the action of local channels of the form $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$. We show that a set of maximally entangled states is by the action of $\mathcal{E}\ensuremath{\bigotimes}\mathcal{I}$ transformed into the set of states that exhibit the same degree of entanglement. Moreover, this degree represents an upper bound on entanglement that is available at the output of the channel irrespective of what the input state of the composite system is. We show that within this bound the entanglement-induced state ordering is ``relative'' and can be changed by the action of local channels. That is, two states ${\ensuremath{\varrho}}_{1}^{(\mathrm{in})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{in})}$ such that the entanglement $E[{\ensuremath{\varrho}}_{1}^{(\mathrm{in})}]$ of the first state is larger than the entanglement $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{in})}]$ of the second state are transformed into states ${\ensuremath{\varrho}}_{1}^{(\mathrm{out})}$ and ${\ensuremath{\varrho}}_{2}^{(\mathrm{out})}$ such that $E[{\ensuremath{\varrho}}_{2}^{(\mathrm{out})}]>E[{\ensuremath{\varrho}}_{1}^{(\mathrm{out})}]$.
We systematically characterize quantum entanglement in all sixteen combinations of two-qubit Bell states across three operational regimes: pure superposition, noiseless mixing, and asymmetric mixing of one noisy and one pure Bell state. For pure superpositions, we calculate the von Neumann entropy and show that entanglement depends critically on the relative phase phi between superposed states. For mixed states, we use concurrence C under three physically motivated decoherence channels. The Depolarizing (DP) and Phase Damping (PD) channels are modeled as global, collective operations on the two-qubit Hilbert space, while Amplitude Damping (AD) is treated as independent local decay on each qubit, reflecting the asymmetric, energy-dissipating nature of spontaneous emission. A central and counterintuitive result is that increasing noise can raise concurrence in specific parameter regimes across all three channels. We further map the boundary between mathematical entanglement (C > 0) and operational quantum non-locality using the Horodecki criterion, finding this boundary to be strongly channel-dependent. Under Phase Damping, the Bell-local entanglement regime vanishes entirely for identical and same-bases mixtures, meaning any surviving entanglement guarantees a CHSH violation. Under Depolarizing noise, a large Bell-local region requires mixing probability r > 1/sqrt(2) to recover non-locality at maximum noise. For Amplitude Damping, certain mixtures exhibit a non-monotonic Horodecki parameter M(T): concurrence decreases monotonically with noise, yet the capacity for CHSH violation is lost at intermediate noise levels and restored at high noise, as maximal damping repurifies the noisy branch. These results provide a unified analytical reference for entanglement and non-locality management across Bell-state combinations and decoherence channels relevant to NISQ architectures.
For a two-qubit system under local depolarizing channels, the most robust and most fragile states are derived for a given concurrence or negativity. For the one-sided channel, the pure states are proved to be the most robust ones, with the aid of the evolution equation for entanglement given by Konrad et al. [Nat. Phys. 4, 99 (2008)]. Based on a generalization of the evolution equation for entanglement, we classify the ansatz states in our investigation by the amount of robustness, and consequently derive the most fragile states. For the two-sided channel, the pure states are the most robust for a fixed concurrence. Under the uniform channel, the most fragile states have the minimal negativity when the concurrence is given in the region [1/2,1]. For a given negativity, the most robust states are the ones with the maximal concurrence, and the most fragile ones are the pure states with minimum of concurrence. When the entanglement approaches zero, the most fragile states under general nonuniform channels tend to the ones in the uniform channel. Influences on robustness by entanglement, degree of mixture, and asymmetry between the two qubits are discussed through numerical calculations. It turns out that the concurrence and negativity are major factors for the robustness. When they are fixed, the impact of the mixedness becomes obvious. In the nonuniform channels, the most fragile states are closely correlated with the asymmetry, while the most robust ones with the degree of mixture.
Using simple models of radiation-matter interaction we study the reduced dynamics of a quantum central system formed by two qubits under the spectator configuration. Purity and concurrence as measures of decoherence and entanglement are evaluated in analytical form. Interestingly in these systems, entanglement sudden death is found despite having less complex dynamics than models used in previous studies. Furthermore, we have analysed the associated local quantum operations generated by the dynamics and how their action on an initial Bell state is characterized in the concurrence-purity diagram.
For a two-qubit system under local depolarizing channels, the most robust and most fragile states are derived for a given concurrence or negativity. For the one-sided channel, the pure states are proved to be the most robust ones, with the aid of the evolution equation for entanglement given by Konrad \emph{et al.} [Nat. Phys. 4, 99 (2008)]. Based on a generalization of the evolution equation for entanglement, we classify the ansatz states in our investigation by the amount of robustness, and consequently derive the most fragile states. For the two-sided channel, the pure states are the most robust for a fixed concurrence. Under the uniform channel, the most fragile states have the minimal negativity when the concurrence is given in the region [1/2,1][1/2,1][1/2,1]. For a given negativity, the most robust states are the ones with the maximal concurrence, and the most fragile ones are the pure states with minimum of concurrence. When the entanglement approaches zero, the most fragile states under general nonuniform channels tend to the ones in the uniform channel. Influences on robustness by entanglement, degree of mixture, and asymmetry between the two qubits are discussed through numerical calculations. It turns out that the concurrence and negativity are major factors for the robustness. When they are fixed, the impact of the mixedness becomes obvious. In the nonuniform channels, the most fragile states are closely correlated with the asymmetry, while the most robust ones with the degree of mixture.