Noisy evolution of graph-state entanglement

A general method for the study of the entanglement evolution of graph states under the action of Pauli maps was recently proposed in Cavalcanti et al. [Phys. Rev. Lett. 103, 030502 (2009)]. This method is based on lower and upper bounds to the entanglement of the entire state as a function only of the state of a (typically) considerably smaller subsystem undergoing an effective noise process related to the original map. This provides a huge decrease in the size of the matrices involved in the calculation of entanglement in these systems. In the present paper we elaborate on this method in detail and generalize it to other natural situations not described by Pauli maps. Specifically, for Pauli maps we introduce an explicit formula for the characterization of the resulting effective noise. Beyond Pauli maps, we show that the same ideas can be applied to the case of thermal reservoirs at arbitrary temperature. In the latter case, we discuss how to optimize the bounds as a function of the noise strength. We illustrate our ideas with explicit exemplary results for several different graphs and particular decoherence processes. The limitations of the method are also discussed.

Figure 1

Mathematical graph associated with a given physical graph state. An exemplary bipartition divides the system into two subparts: the yellow and white regions. The edges in black are the boundary-crossing edges $\mathcal{X}$ and the nodes (also in black) connected by them are the boundary nodes $\mathcal{Y}$. Together they compose the boundary subgraph $G(\mathcal{Y},\mathcal{X})$. The remaining vertices, painted in orange, constitute the nonboundary subsystem.

Figure 2

Same graph as in Fig. 1 but where all non-boundary-crossing edges have been erased, representing the fact that the boundary and nonboundary subsystems are fully unentangled. The entanglement in the whole system is obtained via a calculation involving only the smaller boundary subsystem.

Figure 3

Negativity versus $p$ for 14-qubit (green triangles) and 12-qubit (pink solid curve) cluster states undergoing independent depolarizing noise, and for the bipartition shown in inset. Parameter $p$ can be thought as a parametrization of time (see text).

Figure 4

Negativity versus $p$ for a four-qubit linear cluster subjected to the amplitude damping channel and for the partition displayed in the inset. Solid (upper) black curve, exact entanglement; solid (lower) blue curve, lowest lower bound LLB (obtained by tracing out the flags); dashed-dotted blue curve, upper bound (obtained by convexity); dotted red curve, LB(0) (obtained by measuring the flags in the $Z$ basis); dashed brown curve, LB($\pi /4$) (obtained by measuring the flags in the $X$ basis).

Figure 5

Lower bound LB($\theta$) to the negativity as a function of the angle $\theta$ in the measurement basis, for the same situation as in Fig. 4, and for fixed values of $p$. Each value $p=0.01,0.1,0.3,0.5$ has two curves associated with it. The horizontal (gray) straight line represents the exact entanglement at each $p$, while the blue (black) curve represents the bound LB($\theta$) at this $p$. The red line (vertical) corresponds to $\theta =\frac{\pi }{4}$ (i.e., measurements in the basis $\{|+\rangle ,|-\rangle \}$).

Figure 6

Same as Fig. 5 for $p=0.9$.

Figure 7

Negativity versus $p$ for a four-qubit linear cluster for the bipartition shown in the inset, subjected to the generalized amplitude damping channel in the diffusive limit $\overline{n}\to \infty$. The central curve corresponds to both the exact entanglement (solid black) and LB($\pi /4$) (dashed brown), which coincide exactly. The lower curve corresponds to both LBB (solid blue) and LB(0) (dotted red), which also coincide exactly. The upper curve is the upper bound (22) (dot-dashed blue).

Figure 8

Lower bound LB($\theta$) to the negativity as a function of the angle $\theta$ in the measurement basis, for the same situation as in Fig. 7, and for fixed values of $p$. Each value $p=0.01,0.1,0.3,0.5$ has two curves associated with it. Again, the horizontal (gray) straight line represents the exact entanglement at each $p$, while the blue (black) curve represents the bounds LB($\theta$) at this $p$. The red line (vertical) corresponds to $\theta =\frac{\pi }{4}$ (i.e., measurement in the basis $\{|+\rangle ,|-\rangle \}$).