Entanglement versus mixedness for coupled qubits under a phase damping channel

Quantum entanglement is an essential resource for many quantum communication protocols, such as teleportation [1,2] and dense coding [3,4], allowing an information processing efficiency, which is otherwise unattainable through classical protocols. As such the efficiency of those quantum protocols relies on the ability to isolate the encoding system, being strongly decreased when the system is coupled to an external environment, attaining thus the classical limits, when the quantum system is found in a separable state [5]. The relation of entanglement against separability for mixed entangled states has generated a considerable literature, with many proposals for quantifying it (see for example [6–10] and references therein). Bipartite quantum systems with Hilbert space of dimension 2 ⊗ 2 (coupled qubits) have been exhaustively investigated in order to achieve a precise quantification of entanglement against mixing. Particularly a valuable necessary and sufficient condition for separability of coupled qubits has been given by Peres [7] and Horodecki [8] in terms of the positivity of the partial transpose of the system density matrix, which sets a boundary for comparison to many proposed entanglement measures, such as entanglement of formation, relative entropy of distillation, and relative entropy of entanglement. A very useful paper has appeared recently [9] relating the ordering of many entanglement measures in relation to the degree of mixedness. This paper reinforce and extends the discussion presented in Ref. [6] on maximally entangled mixed states (MEMS), which are states that for a given mixedness achieve the greatest possible entanglement. More recently the imbalance between the sensitivity of common state measures, such as fidelity trace distances, concurrence, tangle and von Neumann entropies when acted on by a depolarizing channel have also been investigated [10]. It was noticed that the size of the imbalance depends intrinsically of the state tangle and of the state purity. The results of Refs. [9,10] were derived for arbitrary entangled mixed states randomly generated bipartite matrices respecting the structure of positive semidefinite operators. An actual bipartite interacting system has its entanglement constrained by its dynamics and thus (for certain given initial states) they never reach the discussed MEMS. The system’s dynamics constrains the degree of entanglement and mixedness to a bounded range. Only the MEMS dynamically connected to the system state are important for setting reference states, and thus only those states are valid for entanglement quantification in terms of distance measures. Those MEMS can be computed by a certain combination of the reduced density matrix of the coupled qubits. It is thus of central importance to analyze the amount of entanglement against mixing present in a quantum system due to a process of deterministic entanglement formation [11] in a noisy channel. In this paper we analyze the degree of entanglement against mixing for a dynamical system composed of two coupled qubits under the phase damping channel. While amplitude damping is certainly the most important source of noise for light field states qubit encoding, the phase damping model describes more appropriately noise over an encoding system composed of internal atomic (ionic) states or even for internal quantum dots states [12–16]. The phase damping channel is particularly interesting in analyzing the degree of entanglement against mixing because it truly induces decoherence without amplitude relaxation effects [5]. We compare many entanglement measures as a function of the joint state purity and discuss how do they relate to each other for the specific dynamical system considered. More specifically we compare concurrence and negativity with Bures distance entanglement measures. While concurrence and negativity are able to quantify the amount of entanglement present in a mixed state, they are not able to distinguish states. The Bures distance entanglement measure, on the other hand, is able to distinguish states and thus can be used to define a ordering of entanglement measures. In its definition however, a deep analysis