Entanglement-breaking channels and entanglement sudden death

The occurrence of entanglement sudden death in the evolution of a bipartite system depends on both the initial state and the channel responsible for the evolution. An extreme case is that of entanglement-breaking channels, which are channels that, acting on only one of the subsystems, drives them to full disentanglement, regardless of the initial state. In general, one can find certain combinations of initial states and channels acting on one or both subsystems that can result in entanglement sudden death or not. Neither the channel nor the initial state are responsible for this effect but their combination. In this paper we show that, for two entangled qubits, when entanglement sudden death occurs, the evolution can be mapped to that of an effective entanglement-breaking channel on a modified initial state. Our results allow to anticipate which states will suffer entanglement sudden death or not for a given evolution. An experiment with polarization-entangled photons demonstrates the utility of this result in a variety of cases.

Figure 1

The relation between channels and states. (a) Any trace-preserving channel corresponds to a bipartite state. (b) However an arbitrary bipartite state can correspond to two nonequivalent channels as illustrated here by means of state preparations departing from two nonequivalent pure states.

Figure 2

Experimental setup. The entangled spontaneous parametric down-conversion (SPDC) pair source is optimized using temporal and spatial compensating birefringent crystals. Bob's polarization qubit can be coupled to a local environment encoded on the path qubit (see the text). The light-green-dashed half-wave plate (HWP) placed on Bob's path is occasionally used to generate state ${\rho }_2$ to simulate the noisy channel on Alice's side.

Figure 3

Measured matrix elements of the Kraus operators reconstructed from quantum process tomography: (a) (top row) for the amplitude damping channel ${\mathcal{E}}_g$ acting on one qubit; (b) (bottom row) for the measured effective channel ${\mathcal{E}}_g\mathbf{\Gamma }$, acting on an initial state with $\omega =0.25$ and $h=0.5$. In both cases the damping parameter is $g=0.6$. Values for matrix elements above 0.20 are written on top of the bars. The height of each bar also codes the magnitude of each element according to the color map to the right.

Figure 4

Experimental results and theoretical predictions. (a) Evolution of the concurrence of states initially characterized by $\omega =0.12$ and different degrees of mixture $h$ under the interaction with a local ADC channel. Concurrence values are coded on the grayscale bar: The continuous map is the solution of Eq. (5), whereas the symbols plot the measured concurrence for different experimental initial conditions and evolution parameter $g$. The inset: measured and theoretical concurrence evolutions of a particular initial state ($h=0.21$) showing ESD. Vertical error bars were calculated by Monte Carlo simulation of experimental runs with the same Poissonian statistics. (b) Parameter regions that define the quantum map (13) as an EBC or non-EBC: Any pair of values $h,\omega$ that falls into the upper left corner generates an EBC. The blue circles are initial conditions for which we measured concurrence values of zero for some value of $g<1$, whereas the solid black line represents the theoretical boundary between EB and non-EB channels. The shaded area represents the region of experimental error. (c) Evolution of the concurrence of $X$ states characterized by their initial mixture $h$ under the action of local phase damping channels. Contour lines are plotted in (a) and (c) as references for certain concurrence values, whereas crossed-out circles indicate zero-concurrence experimental outcomes.