Decoherence and entanglement dynamics in fluctuating fields

We study pure phase damping of two qubits due to fluctuating fields. As frequently employed, decoherence is thus described in terms of random unitary (RU) dynamics, that is, a convex mixture of unitary transformations. Based on a separation of the dynamics into an average Hamiltonian and a noise channel, we are able to analytically determine the evolution of both entanglement and purity. This enables us to characterize the dynamics in a concurrence-purity (CP) diagram: We find that RU phase-damping dynamics sets constraints on accessible regions in the CP plane. We show that initial state and dynamics contribute to final entanglement independently.

Figure 1

The CP diagram. The light gray area shows the region of all physical states of two qubits, bounded by the maximally entangled mixed states ($C_{\mathrm{MEMS}}$). The possible $C(P)$ relations for a Bell state subject to single-sided unital dynamics or, equivalently, of all doubly chaotic states, is highlighted in blue (dark gray). This region is bounded by the curves valid for Werner states $C_{\mathrm{W}}$ and for Bell states under single-sided phase damping $C_{\mathrm{D}}$, respectively. The diagram also reveals the maxium $C(P)$ relation of doubly chaotic states with Bell rank $3$, $C_3(P)$ (dashed line).

Figure 2

(a) For a pure separable initial state subject to pure two-qubit phase-damping, the maximally achievable concurrence is bounded from above by $C_{\mathrm{D}}$. The corresponding forbidden area is highlighted in red (dark gray). The blue (dashed) line represents the evolution under a one-parameter class of phase damping channels (see text for details). (b) The accessible value of concurrence depends on expectation value $\langle {\sigma }_z^{(i)}\rangle$ evaluated for either one of the qubits $i=A,B$. Only for $\langle {\sigma }_z^A\rangle =\langle {\sigma }_z^B\rangle =0$ may the relation $C_{\mathrm{D}}$ (thick red line) be obtained.

Figure 3

(a) In case of general uncorrelated phase-damping, the maximally achievable concurrence is identical to the value given by the Werner states, $C_{\mathrm{W}}$. As before, the forbidden area is highlighted in red (dark gray). The role of correlations may be observed in (b), where both bounds are violated. In both diagrams, an exemplary evolution under a one-parameter class of phase-damping channels is shown (dashed blue lines).

Figure 4

Accessible region for a single-sided phase-damping channel for states of varying initial entanglement $C_0=0.2$ (0.4, 0.6, 0.8, 1). For $C_0=1$, the lower and upper bounds are equal, given by the known curve $C_{\mathrm{D}}$.

Figure 5

For pure two-qubit phase damping we can identify states with robust entanglement. The plot is obtained for an initial state parameterized by $\chi =0$, ${\theta }_1=3\pi /16$, ${\theta }_2=\pi /8$ (see text), leading to robust entanglement for $C_0⩽C_{>}\approx 0.5142$ (dashed red line). The blue (thick) lines show exemplary evolution of pure states with initial concurrence $C_0=1.0$ (0.8, 0.6, 0.4, 0.2).